diff --git a/工具v2/批量收录题目.py b/工具v2/批量收录题目.py
index b5d4bfb5..3e9a3ec2 100644
--- a/工具v2/批量收录题目.py
+++ b/工具v2/批量收录题目.py
@@ -1,5 +1,5 @@
 #修改起始id,出处,文件名
-starting_id = 18576 #起始id设置, 来自"寻找空闲题号"功能
+starting_id = 18663 #起始id设置, 来自"寻找空闲题号"功能
 raworigin = "" #题目来源的前缀(中缀在.tex文件中)
 filename = r"C:\Users\wangweiye\Documents\wwy sync\临时工作区\空中课堂必修第二册例题与习题.tex" #题目的来源.tex文件
 editor = "王伟叶" #编辑者姓名
diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json
index c5684959..441fa941 100644
--- a/题库0.3/Problems.json
+++ b/题库0.3/Problems.json
@@ -480263,6 +480263,1370 @@
         "space": "4em",
         "unrelated": []
     },
+    "018663": {
+        "id": "018663",
+        "content": "如图, 已知斜三棱柱$ABC-A' B' C'$的底面是正三角形, 侧棱$AA' \\perp BC$, 并且与底面所成角是$60^{\\circ}$. 设侧棱长为$l$, 点$A'$在平面$ABC$上的射影为点$O$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\h{2}\n\\draw ({-\\l/2},0,0) node [left] {$A$} coordinate (A);\n\\draw (0,0,{\\l/2*sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ({\\l/2},0,0) node [right] {$C$} coordinate (C);\n\\draw (A) ++ ({sqrt(3)/2},{sqrt(3)},{1/2}) node [left] {$A'$} coordinate (A');\n\\draw (B) ++ ({sqrt(3)/2},{sqrt(3)},{1/2}) node [below right] {$B'$} coordinate (B');\n\\draw (C) ++ ({sqrt(3)/2},{sqrt(3)},{1/2}) node [right] {$C'$} coordinate (C');\n\\draw (A) -- (B) -- (C) (A) -- (A') (B) -- (B') (C) -- (C') (A') -- (B') -- (C') (A') -- (C');\n\\draw [dashed] (A) -- (C);\n\\draw (A) ++ ({sqrt(3)/2},0,{1/2}) node [above right] {$O$} coordinate (O);\n\\draw [dashed] (A')--(O)(A)--($(B)!0.5!(C)$);\n\\end{tikzpicture}\n\\end{center}\n(1) 求此三棱柱的高;\\\\\n(2) 求证: 侧面$BB' C' C$为矩形;\\\\\n(3) 求证: 点$O$在$\\angle BAC$的平分线上.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018664": {
+        "id": "018664",
+        "content": "证明: (1) 过圆柱的轴的任意平面与圆柱形成的截面都是全等的矩形;\\\\\n(2) 任一平行于圆柱底面的平面与圆柱形成的截面都是与底面全等的圆.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018665": {
+        "id": "018665",
+        "content": "将一个长和宽分别为$4$和 $3$ 的矩形(及其内部)绕其一条边所在的直线旋转一周后得到圆柱, 过该圆柱的轴的平面与该圆柱形成的截面面积为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018666": {
+        "id": "018666",
+        "content": "在各条棱长均为$1$的正三棱柱$ABC-A_1B_1C_1$中, 求$AC_1$与平面$ABB_1A_1$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018667": {
+        "id": "018667",
+        "content": "下列命题中真命题为\\blank{50}. (请填入全部正确的序号)\\\\\n\\textcircled{1} 棱柱的侧棱长都相等, 侧面都是平行四边形;\\\\\n\\textcircled{2} 有两个侧面是矩形的棱柱是直棱柱;\\\\\n\\textcircled{3} 有一个侧面与底面垂直的棱柱是直棱柱.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018668": {
+        "id": "018668",
+        "content": "已知圆柱的母线$AA_1$的长为 $4$, 底面半径为 $3$ . 其中$O$、$O_1$分别为该圆柱上、下底面的圆心, $OA$与$O_1B_1$分别为两个底面的半径, 且直线$OA$与$O_1B_1$所成角的大小为$\\dfrac{\\pi}{3}$, 求直线$AB_1$与$OO_1$所成角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018669": {
+        "id": "018669",
+        "content": "如图, 已知三棱柱的底面三角形$ABC$的三边长分别是$AB=13 \\text{cm}$, $BC=5 \\text{cm}$, $CA=12 \\text{cm}$, 侧棱$AA'=20 \\text{cm}$, 且侧棱$AA'$与底面所成的角为$60^{\\circ}$. 求这个三棱柱的体积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.15]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (13,0,0) node [right] {$B$} coordinate (B);\n\\draw ({144/13},0,{-60/13}) node [above left] {$C$} coordinate (C);\n\\draw (A) ++ (10,{10*sqrt(3)},0) node [left] {$A'$} coordinate (A');\n\\draw (B) ++ (10,{10*sqrt(3)},0) node [right] {$B'$} coordinate (B');\n\\draw (C) ++ (10,{10*sqrt(3)},0) node [above] {$C'$} coordinate (C');\n\\draw (A)--(B)--(B')--(C')--(A')--cycle(A')--(B');\n\\draw [dashed] (A)--(C)--(B)(C)--(C');\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018670": {
+        "id": "018670",
+        "content": "某工厂准备采购一批密度为$7.2 \\mathrm{g} / \\text{cm}^3$的铸铁, 用于浇铸加工成$100$根空心铸铁直管, 其底面是圆环, 外直径为$10 \\text{cm}$, 内直径为$9 \\text{cm}$, 每根空心铸铁直管的高度为$1500 \\text{cm}$. 假设浇铸加工过程中不计损耗, 那么该工厂此次至少需采购铸铁多少千克? (结果精确到$1$千克)",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018671": {
+        "id": "018671",
+        "content": "有一块如图所示的正方体木块, 若经过棱$A_1B_1$上的一点$E$和棱$BC$将木块锯开, 所得到的两个木块的体积分别为$800 \\text{cm}^3$、$200 \\text{cm}^3$, 请确定点$E$的位置.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\filldraw ($(B_1)!0.4!(A_1)$) circle (0.03) node [above] {$E$} coordinate (E);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018672": {
+        "id": "018672",
+        "content": "在底面为梯形的四棱柱$ABCD-A_1B_1C_1D_1$中, $AB\\parallel CD$, 侧面$CDD_1C_1$的面积为 $2$, 侧面$ABB_1A_1$的面积为 $4$, 若侧面$CDD_1C_1$与侧面$ABB_1A_1$之间的距离为 $3$, 则该四棱柱的体积为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018673": {
+        "id": "018673",
+        "content": "如图, 在斜四棱柱$ABCD-A_1B_1C_1D_1$中, 已知底面$ABCD$为矩形, $\\angle A_1AB=\\angle A_1AD=60^{\\circ}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\def\\l{5}\n\\def\\m{4}\n\\def\\n{3}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [below] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ ({3/2},{3/sqrt(2)},{-3/2}) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ ({3/2},{3/sqrt(2)},{-3/2}) node [above] {$B_1$} coordinate (B_1);\n\\draw (C) ++ ({3/2},{3/sqrt(2)},{-3/2}) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ ({3/2},{3/sqrt(2)},{-3/2}) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw (A) ++ ({3/2},0,{-3/2}) circle (0.03) node [right] {$O$} coordinate (O);\n\\draw [dashed] (A_1)--(O);\n\\end{tikzpicture}\n\\end{center}    \n(1) 求证: 点$A_1$在底面$ABCD$上的射影$O$在$\\angle BAD$的平分线上;\\\\\n(2) 若$AB=5, AD=4, AA_1=3$, 求该斜四棱柱的体积.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018674": {
+        "id": "018674",
+        "content": "已知正六棱柱的底面边长为$4 \\text{cm}$, 高为$16 \\text{cm}$, 求正六棱柱的表面积.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018675": {
+        "id": "018675",
+        "content": "已知圆柱的底面半径为$3 \\text{cm}$, 高为$12 \\text{cm}$, 求圆柱的表面积.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018676": {
+        "id": "018676",
+        "content": "已知正六棱柱的底面边长为$4 \\text{cm}$, 高为$16 \\text{cm}$, 圆柱的底面半径为$3 \\text{cm}$, 高为$12 \\text{cm}$, 若将圆柱放在正六棱柱上, 圆柱的下底面的中心与正六棱柱的上底面的中心重合, 组成一个新的几何体, 求该几何体的表面积.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018677": {
+        "id": "018677",
+        "content": "一张 A4 纸的规格为: $210 \\mathrm{mm} \\times 297 \\mathrm{mm}$, 把它作为一个圆柱的侧面, 则卷成的圆柱体体积约为多少? (结果精确到$1 \\mathrm{mm}^3$)",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018678": {
+        "id": "018678",
+        "content": "一张白纸的长为 $10$, 宽为 $4$, 把它折起来, 作为一个长方体的四个侧面. 求当底面边长分别为多少时, 长方体的体积最大?",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018679": {
+        "id": "018679",
+        "content": "如图, 已知直四棱柱$ABCD-A_1B_1C_1D_1$的底面是菱形, 截面$A_1ACC_1$与截面$B_1BDD_1$的面积分别是$S_1$和$S_2$, 求它的侧面积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1.5,0,0) node [left] {$A$} coordinate (A);\n\\draw (1.5,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,0,1) node [below] {$B$} coordinate (B);\n\\draw (0,0,-1) node [above right] {$D$} coordinate (D);\n\\foreach \\i/\\j in {A/left,B/below left,C/right,D/above}\n{\\draw (\\i) ++ (0,2,0) node [\\j] {$\\i_1$} coordinate (\\i_1);};\n\\draw (A)--(B)--(C)--(C_1)--(D_1)--(A_1)--cycle(B)--(B_1)(A_1)--(B_1)--(C_1)(B_1)--(D_1)(A_1)--(C_1);\n\\draw [dashed] (A)--(D)--(C)(A)--(C)(B)--(D)--(D_1);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018680": {
+        "id": "018680",
+        "content": "如图, 已知长方体$ABCD-A_1B_1C_1D_1$的表面积是$22$, 所有棱长的和是$24$, 求$BD_1$的长.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{1.5}\n\\def\\n{1.2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018681": {
+        "id": "018681",
+        "content": "已知圆柱的底面面积是$S$, 侧面展开图是一个正方形, 求该圆柱的表面积.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018682": {
+        "id": "018682",
+        "content": "证明: 在正三棱锥中, 任意两条异面的棱都相互垂直.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ($1/3*(A)+1/3*(B)+1/3*(C)$) ++ (0,{sqrt(13)/3},0) node [above] {$P$} coordinate (P);\n\\draw (P)--(A)--(B)--(C)--cycle(P)--(B);\n\\draw [dashed] (A)--(C);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018683": {
+        "id": "018683",
+        "content": "已知正三棱锥$P-ABC$的底面边长为 $6$, 侧棱长为$5$, 求该正三棱锥的高.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (2,0,0) node [right] {$C$} coordinate (C);\n\\draw (1,0,{sqrt(3)}) node [below] {$B$} coordinate (B);\n\\draw ($1/3*(A)+1/3*(B)+1/3*(C)$) ++ (0,{sqrt(13)/3},0) node [above] {$P$} coordinate (P);\n\\draw (P)--(A)--(B)--(C)--cycle(P)--(B);\n\\draw [dashed] (A)--(C);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018684": {
+        "id": "018684",
+        "content": "已知正四棱锥$P-ABCD$的侧棱长与底面边长均为$1$, 求侧棱与底面所成角的大小.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1,0,1) node [left] {$A$} coordinate (A);\n\\draw (1,0,1) node [right] {$B$} coordinate (B);\n\\draw (1,0,-1) node [right] {$C$} coordinate (C);\n\\draw (-1,0,-1) node [below] {$D$} coordinate (D);\n\\draw (0,{sqrt(2)},0) node [above] {$P$} coordinate (P);\n\\draw (P)--(A)--(B)--(C)--cycle(P)--(B);\n\\draw [dashed] (A)--(D)--(C)(D)--(P);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018685": {
+        "id": "018685",
+        "content": "如图, 用平行于圆锥$P-O_1$底面的平面截这个圆锥, 得到一个小圆锥$P-O_2$. 如果这两个圆锥的高分别是$h_1$、$h_2$, 求这两个圆锥的底面面积之比.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (-1,0) arc (180:360:1 and 0.25) -- (0,2) -- (-1,0);\n\\draw (-0.5,1) arc (180:360:0.5 and 0.125);\n\\draw [dashed] (1,0) arc (0:180:1 and 0.25) (0.5,1) arc (0:180:0.5 and 0.125);\n\\draw (0,0) node [left] {$O_1$};\n\\draw (0,1) node [left] {$O_2$};\n\\draw [dashed] (0,0) -- (1,0) (0,1)--(0.5,1) (0,0) -- (0,2);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018686": {
+        "id": "018686",
+        "content": "举几个生活中棱锥、圆锥的实例.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018687": {
+        "id": "018687",
+        "content": "已知正四棱锥的底面边长是$a$, 高是$h$. 求其侧棱长.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018688": {
+        "id": "018688",
+        "content": "已知正六棱锥的底面边长是$4 \\text{cm}$, 侧棱长是$8 \\text{cm}$. 求它的侧面和底面所成的二面角的大小.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018689": {
+        "id": "018689",
+        "content": "已知圆锥底面半径为$r$, 过该圆锥的轴的截面是直角三角形, 求这个截面的面积.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018690": {
+        "id": "018690",
+        "content": "已知三棱锥$P-ABC$的三条侧棱$PA$、$PB$、$PC$两两互相垂直, 底面$ABC$内一点$M$到三个侧面的距离分别为$2$、$3$、$6$, 求线段$PM$的长.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018691": {
+        "id": "018691",
+        "content": "若圆锥的底面面积与过轴的截面面积之比为$\\sqrt{3} \\pi$, 则其母线与底面所成角的大小为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018692": {
+        "id": "018692",
+        "content": "已知一个正四棱台上、下底面的边长分别是$a$和$b$, 高是$h$. 求经过相对的两条侧棱的截面面积.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018693": {
+        "id": "018693",
+        "content": "如图, 设$E$、$F$分别是给定正方体$ABCD-A_1B_1C_1D_1$的棱$C_1D_1$和$CD$上的任意点, 求证: 三棱锥$E-ABF$的体积是定值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw ($(C_1)!0.6!(D_1)$) node [above] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(D)$) node [above left] {$F$} coordinate (F);\n\\draw [dashed] (A)--(E)--(B)(A)--(F)--(B)(E)--(F);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018694": {
+        "id": "018694",
+        "content": "如图, 设正方体$ABCD-A_1B_1C_1D_1$的棱长为$a$, 求顶点$B_1$到面$BA_1C_1$的距离.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\draw (B)--(C_1)--(A_1)--cycle;\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018695": {
+        "id": "018695",
+        "content": "如图, 设台体上、下底面积分别为$S'$和$S$, 上下底面的距离为$h$. 求证: $V_{\\text {台 }}=\\dfrac{1}{3}(S'+\\sqrt{S' S}+S) h$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1,0,1)  coordinate (A);\n\\draw (1,0,1)  coordinate (B);\n\\draw (1,0,-1)  coordinate (C);\n\\draw (-1,0,-1)  coordinate (D);\n\\draw (-0.4,1,0.4) coordinate (A_1);\n\\draw (0.4,1,0.4) coordinate (B_1);\n\\draw (0.4,1,-0.4) coordinate (C_1);\n\\draw (-0.4,1,-0.4) coordinate (D_1);\n\\draw (A)--(B)--(C)--(C_1)--(D_1)--(A_1)--cycle(A_1)--(B_1)--(C_1)(B_1)--(B);\n\\draw [dashed] (A)--(D)--(C)(D)--(D_1);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018696": {
+        "id": "018696",
+        "content": "如图, 将一个长方体沿相邻三个面的对角线截出一个棱锥, 求棱锥的体积与剩下的几何体的体积之比.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{2.3}\n\\draw (0,0,0)  coordinate (A);\n\\draw (A) ++ (\\l,0,0)  coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m)  coordinate (C);\n\\draw (A) ++ (0,0,-\\m)  coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0)  coordinate (A_1);\n\\draw (B) ++ (0,\\n,0)  coordinate (B_1);\n\\draw (C) ++ (0,\\n,0)  coordinate (C_1);\n\\draw (D) ++ (0,\\n,0)  coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\filldraw [pattern = north east lines] (B)--(C_1)--(A_1)--cycle;\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018697": {
+        "id": "018697",
+        "content": "如图, 一个倒立的圆锥形水杯, 底面半径为$10 \\text{cm}$, 高为$15 \\text{cm}$. 将一定量的水注入其中, 水形成的圆锥高为$h \\text{cm}$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw (0,0) --++ (2,3) (0,0) --++ (-2,3);\n\\draw (2,3) arc (0:360:2 and 0.5);\n\\fill [pattern = north east lines] (0,0) -- (1,1.5) arc (0:180:1 and 0.25) -- cycle;\n\\draw [dashed] (1,1.5) arc (0:180:1 and 0.25);\n\\draw (1,1.5) arc (360:180:1 and 0.25);\n\\draw [dashed] (0,0) -- (0,3) -- (2,3)(0,1.5) --++ (1,0);\n\\draw (1.1,1.5) -- (1.7,1.5) (0.1,0) -- (1.7,0);\n\\draw [<->] (1.4,0) -- (1.4,1.5) node [midway, fill=white] {$h$};\n\\end{tikzpicture}\n\\end{center}\n(1) 用$h$表示水的体积;\\\\\n(2) 若水的体积恰为圆锥形水杯容积的一半, 求$h$的值. (结果精确到$0.01 \\text{cm})$.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018698": {
+        "id": "018698",
+        "content": "我国古代数学名著《数书九章》中有``天池盆测雨''题: 在下雨时, 用一个圆台形的天池盆接雨水. 天池盆盆口直径为二尺八寸, 盆底直径为一尺二寸, 盆深一尺八寸. 若盆中积水深九寸, 则降雨量是\\blank{50}寸.(注: \\textcircled{1} 一尺等于十寸; \\textcircled{2} 降雨量等于盆中积水体积除以盆口面积).",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018699": {
+        "id": "018699",
+        "content": "已知正三棱锥$O-ABC$的底面边长为$2 \\text{cm}$, 高为$1 \\text{cm}$. 求该三棱锥的表面积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$A$} coordinate (A);\n\\draw (1,0,{sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw (1,1,{sqrt(3)/3}) node [above] {$O$} coordinate (O);\n\\draw (O)--(A)--(C)--(B)--cycle(O)--(C);\n\\draw [dashed] (A)--(B);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018700": {
+        "id": "018700",
+        "content": "已知正三棱锥$O-ABC$的底面边长为$2 \\text{cm}$, 高为$1 \\text{cm}$. 过棱$OB$的中点$B'$作平行于底面$ABC$的平面$\\alpha$截该正三棱锥, 平面$\\alpha$分别与棱$OC$、$OA$交于$C'$和$A'$. 求棱台$ABC-A' B' C'$的表面积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$A$} coordinate (A);\n\\draw (1,0,{sqrt(3)}) node [below] {$C$} coordinate (C);\n\\draw (1,1,{sqrt(3)/3}) node [above] {$O$} coordinate (O);\n\\draw (O)--(A)--(C)--(B)--cycle(O)--(C);\n\\draw [dashed] (A)--(B);\n\\draw ($(O)!0.5!(A)$) node [right] {$A'$} coordinate (A');\n\\draw ($(O)!0.5!(B)$) node [left] {$B'$} coordinate (B');\n\\draw ($(O)!0.5!(C)$) node [below left] {$C'$} coordinate (C');\n\\draw (A')--(C')--(B');\n\\draw [dashed] (A')--(B');\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018701": {
+        "id": "018701",
+        "content": "如图, 已知$AB$是圆锥的底面直径, 顶点$S$与$AB$组成的三角形是正三角形, 若圆锥的体积为$9 \\sqrt{3} \\pi$, 点$O$为底面的圆心. 求该圆锥的表面积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 1.5]\n\\draw (0,{sqrt(3)}) node [above] {$S$} coordinate (S);\n\\draw (-1,0) node [left] {$A$} coordinate (A);\n\\draw (1,0) node [right] {$B$} coordinate (B);\n\\draw (A)--(S)--(B)arc (360:180:1 and 0.25);\n\\draw (0,0) node [above left] {$O$} coordinate (O);\n\\draw [dashed] (A)--(B)(O)--(S) (B) arc (0:180:1 and 0.25);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018702": {
+        "id": "018702",
+        "content": "有一个圆锥形漏斗, 其底面直径是$10 \\text{cm}$, 母线长为$20 \\text{cm}$. 在漏斗口的点$P$处用一根绳子将漏斗挂在墙面上, 当绳子的长度最短时, 可以紧紧地箍住漏斗, 不会上下滑动. 求此时绳子的长度. (结果精确到$1 \\text{cm}$)",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018703": {
+        "id": "018703",
+        "content": "已知圆锥的母线长为$2$ . 记$S$为侧面积, $V$为体积, 求$\\dfrac{V}{S}$取得最大值时圆锥的体积$V$的值.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018704": {
+        "id": "018704",
+        "content": "平面上的图形由如图所示的三个直角三角形构成, 这三个直角三角形 (及其内部) 绕直线$l$旋转一周形成一个几何体, 该几何体的表面积为体积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.3]\n\\draw (0,0) -- (-2,-3) -- (0,-3) -- (-4,-6) -- (0,-6) -- (-6,-9) -- (0,-9);\n\\draw (0,1) -- (0,-12) node [right] {$l$};\n\\foreach \\i in {-6,-3,0}\n{\\draw (0.1,\\i) -- (1.2,\\i);\n\\draw [<->] (0.8,\\i) --++ (0,-3) node [midway,fill=white] {$3$};};\n\\draw (0.1,-9) -- (1.2,-9);\n\\draw [dashed] (-2,-2) -- (-2,-11) (-4,-5) -- (-4,-11) (-6,-8) -- (-6,-11);\n\\foreach \\i in {0,-2,-4}\n{\\draw [<->] (\\i,-10) --++ (-2,0) node [midway, fill = white] {$2$};};\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018705": {
+        "id": "018705",
+        "content": "观察如图所示的纸篓的几何结构, 可以发现, 它是由许多直线段支撑起来的, 这些密集的直线段给了我们一个旋转面的形象, 请尝试描述此曲面的形成过程.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\foreach \\i in {0,10,...,350}\n{\\draw ({cos(\\i)},-1,{sin(\\i)}) -- ({cos(\\i+90)},1,{sin(\\i+90)});\n\\draw ({cos(\\i)},-1,{sin(\\i)}) -- ({cos(\\i-90)},1,{sin(\\i-90)});};\n\\draw [thick, domain = 0:360] plot ({cos(\\x)},-1,{sin(\\x)}) plot ({cos(\\x)},1,{sin(\\x)});\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018706": {
+        "id": "018706",
+        "content": "已知球的半径为$5$, $OO_1=4$. 求小圆$O_1$的半径.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018707": {
+        "id": "018707",
+        "content": "如图, 设$AB$是球$O$的一条直径, 过球心$O$作一个大圆$ODC$与$AB$垂直, 过直径$AB$上不同于点$O$的任一点$O_1$作与$AB$垂直的平面, 与球$O$交于小圆$O_1$, 过直径$AB$作两个平面与球分别交于两个大圆$OEC$和$OFD$, $E$和$F$分别是这两个大圆的圆周与圆$O_1$的交点. 求证:\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 2]\n\\draw (0,0) node [left] {$O$} coordinate (O);\n\\draw (O) circle (1);\n\\draw (1,0) node [right] {$C$} coordinate (C);\n\\draw ({sqrt(3)/2},{1/2}) node [right] {$E$} coordinate (E);\n\\draw (0,1) node [above] {$A$} coordinate (A);\n\\draw (0,-1) node [below] {$B$} coordinate (B);\n\\path [name path=AB, draw] (B) arc (270:90:{1/3} and 1);\n\\path [name path=CD, draw] (C) arc (360:180:1 and {1/3});\n\\path [name intersections = {of = AB and CD, by = D}] (D) node [below left] {$D$};\n\\path [name path = EF, draw] (E) arc (360:180:{sqrt(3)/2} and {sqrt(3)/6});\n\\path [name intersections = {of = AB and EF, by = F}] (F) node [below left] {$F$};\n\\draw [dashed] (E) arc (0:180:{sqrt(3)/2} and {sqrt(3)/6}) (C) arc (0:180:1 and {1/3}) (B) arc (270:450:{1/3} and 1);\n\\draw (0,{1/2}) node [left] {$O_1$} coordinate (O_1);\n\\draw [dashed] (F)--(O_1)--(E) (D)--(O)--(C)(A)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) $\\angle DOC$、$\\angle FO_1E$都是二面角$D-AB-C$的平面角;\\\\\n(2) $OE$和$OF$与平面$ODC$所成的角相等.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018708": {
+        "id": "018708",
+        "content": "若用一平面截球所得的小圆半径为$6$, 球的半径为$10$, 则球心到该平面的距离为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018709": {
+        "id": "018709",
+        "content": "设地球的半径为$R$, 若$A$在北纬$30^{\\circ}$的纬线圈上, 则此纬线圈构成的小圆面积为\\blank{50}. (结果用$R$表示)",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018710": {
+        "id": "018710",
+        "content": "已知球$O$, $A$为球面上一点, 若过$OA$的中点$M$且垂直于$OA$的平面截球所得的小圆面积为$3 \\pi$, 则球$O$的半径为\\blank{50}.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018711": {
+        "id": "018711",
+        "content": "已知球的半径为$25$, 有两个平行平面截球所得的截面面积分别为$49 \\pi$和$400 \\pi$, 求这两个平行平面间的距离.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018712": {
+        "id": "018712",
+        "content": "设地球的半径是$R$, 如图, 在北纬$45^{\\circ}$的纬线上有$A, B$两点, $O_1$为小圆圆心.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 2]\n\\filldraw (0,0) circle (0.015) node [left] {$O$} coordinate (O);\n\\filldraw (0,{sqrt(2)/2}) circle (0.015) node [above left] {$O_1$} coordinate (O_1);\n\\draw (O) circle (1);\n\\draw (1,0) arc (360:180:1 and {1/4}) (45:1) arc (360:180:{sqrt(2)/2} and {sqrt(2)/8});\n\\draw [dashed] (1,0) arc (0:180:1 and {1/4}) (45:1) arc (0:180:{sqrt(2)/2} and {sqrt(2)/8});\n\\draw (O_1) ++ (-40:{sqrt(2)/2} and {sqrt(2)/8}) node [below] {$B$} coordinate (B);\n\\draw (O_1) ++ (-130:{sqrt(2)/2} and {sqrt(2)/8}) node [below] {$A$} coordinate (A);\n\\draw [dashed] (A)--(O_1)--(B);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$O_1B$的长;\\\\\n(2) 若$\\angle AO_1B=90^{\\circ}$, 在过$A$、$B$的大圆$OAB$中, 求弦$AB$所对的劣弧长.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018713": {
+        "id": "018713",
+        "content": "已知$A$、$B$是球$O$的球面上两点, $\\angle AOB=90^{\\circ}$, $C$为该球面上的动点. 若三棱锥$O-ABC$体积的最大值为$36$, 则球$O$的半径为\\bracket{20}.\n\\fourch{$2$}{$4$}{$6$}{$8$}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018714": {
+        "id": "018714",
+        "content": "有一种空心钢球, 质量为$142 \\mathrm{g}$, 测得球的外直径等于$5.0 \\text{cm}$. 求它的内直径. (钢的密度是$7.9 \\mathrm{g} / \\text{cm}^3$, 结果精确到$0.1 \\text{cm}$)",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018715": {
+        "id": "018715",
+        "content": "已知圆柱的底面直径与高都等于球的直径. 求证:\\\\\n(1) 球的表面积等于圆柱的侧面积;\\\\\n(2) 球的表面积等于圆柱表面积的$\\dfrac{2}{3}$.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018716": {
+        "id": "018716",
+        "content": "已知圆柱的底面直径与高都等于球的直径. 求球的体积与圆柱体积的比值.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018717": {
+        "id": "018717",
+        "content": "半正多面体亦称``阿基米德多面体'', 是由边数不全相同的正多边形围成的多面体, 体现了数学的对称美. 以正方体每条棱的中点为顶点构造一个半正多面体, 如图, 它由八个正三角形和六个正方形构成, 若它的所有棱长都为$1$, 其各顶点都位于球$O$的球面上, 则球$O$的表面积为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(220:0.5cm)}]\n\\draw (-1,0,0)  coordinate (A);\n\\draw (0,0,1)  coordinate (B);\n\\draw (1,0,0)  coordinate (C);\n\\draw (0,0,-1)  coordinate (D);\n\\draw (-1,1,1)  coordinate (E);\n\\draw (1,1,1)  coordinate (F);\n\\draw (1,1,-1) coordinate (G);\n\\draw (-1,1,-1)  coordinate (H);\n\\draw (-1,2,0)  coordinate (M);\n\\draw (0,2,1)  coordinate (N);\n\\draw (1,2,0)  coordinate (P);\n\\draw (0,2,-1)  coordinate (Q);\n\\draw (A)--(B)--(E)--cycle (B)--(C)--(F)--cycle (C)--(G) (G)--(P) (F)--(P)--(N) --cycle (M)--(N)--(E)--cycle (M)--(Q)--(P);\n\\draw [dashed] (A)--(D)--(H)--cycle (C)--(D)--(G) (M)--(H)--(Q)--(G);\n\\draw [dotted] (-1,0,1) -- (1,0,1) -- (1,0,-1) -- (-1,0,-1) -- cycle;\n\\draw [dotted] (-1,2,1) -- (1,2,1) -- (1,2,-1) -- (-1,2,-1) -- cycle;\n\\foreach \\i in {(-1,0,1),(1,0,1),(1,0,-1),(-1,0,-1)}\n{\\draw [dotted] \\i --++ (0,2,0);};\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018718": {
+        "id": "018718",
+        "content": "已知三棱锥$P-ABC$的四个顶点都在球$O$的球面上, $AB=2$, $AC=\\sqrt{7}$, $\\tan \\angle BAC=\\dfrac{\\sqrt{3}}{2}$, $PA=\\sqrt{2}$, 当此三棱锥的体积最大时, 求球$O$的体积.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018719": {
+        "id": "018719",
+        "content": "如图, 在斜三棱柱$ABC-A_1B_1C_1$中, $\\angle A_1AC=\\angle ACB=\\dfrac{\\pi}{2}$, $\\angle AA_1C=\\dfrac{\\pi}{6}$, 侧棱$BB_1$与底面所成的角为$\\dfrac{\\pi}{3}$, $AA_1=4 \\sqrt{3}$, $BC=4$. 求斜三棱柱$ABC-A_1B_1C_1$的体积$V$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw ({-2*sqrt(2)},0,0) node [below] {$A$} coordinate (A);\n\\draw ({2*sqrt(2)},0,0) node [below] {$B$} coordinate (B);\n\\draw (0,0,{-2*sqrt(2)}) node [above right] {$C$} coordinate (C);\n\\draw (A) -- (B);\n\\draw [dashed] (A) -- (C) -- (B);\n\\draw (A) ++ ({-sqrt(6)},6,{-sqrt(6)}) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ ({-sqrt(6)},6,{-sqrt(6)}) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ ({-sqrt(6)},6,{-sqrt(6)}) node [above] {$C_1$} coordinate (C1);\n\\draw [dashed]  (C) -- (C1) (A1) -- (C);\n\\draw (A) -- (A1) (B) -- (B1) (A1) -- (B1) (A1) -- (C1) -- (B1); \n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018720": {
+        "id": "018720",
+        "content": "如图, 圆锥$P-O$的底面直径和高均是$a$, 过$PO$的中点$O'$作平行于底面的截面, 以该截面为底面挖去一个圆柱, 求剩下几何体的体积和表面积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0) node [left] {$O$} coordinate (O);\n\\draw (0,2) node [above] {$P$} coordinate (P);\n\\draw (0,1) node [left] {$O'$} coordinate (O1);\n\\draw (P) -- (-1,0) (P) -- (1,0);\n\\draw [dashed] (P) --++ (0,-2) --++ (1,0);\n\\draw [dashed] (-1,0) arc (180:0:1 and 0.25) (-0.5,1) arc (180:0:0.5 and 0.125);\n\\draw (-1,0) arc (180:360:1 and 0.25);\n\\draw [dashed] (-0.5,0) --++ (0,1) (0.5,0) --++ (0,1);\n\\draw (-0.5,1) arc (180:360:0.5 and 0.125);\n\\draw [dashed] (O) ellipse (0.5 and 0.125);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018721": {
+        "id": "018721",
+        "content": "如图, 三棱锥$P-MNQ$中, $PM \\perp NQ$, $PM \\perp MN$, $NQ \\perp MN$. 若$MN=NQ=1$, 二面角$P-NQ-M$的大小为$\\dfrac{\\pi}{4}$, 求:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$M$} coordinate (M);\n\\draw (1,0,1) node [below] {$N$} coordinate (N);\n\\draw (2,0,0) node [right] {$Q$} coordinate (Q);\n\\draw (0,{sqrt(2)},0) node [above] {$P$} coordinate (P);\n\\draw (P)--(M)--(N)--(Q)--cycle(P)--(N);\n\\draw [dashed] (M)--(Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 三棱锥$P-MNQ$的体积;\\\\\n(2) 点$M$到平面$PNQ$的距离.",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018722": {
+        "id": "018722",
+        "content": "如图, 三棱锥$P-MNQ$中, $PM \\perp NQ$, $PM \\perp MN$, $NQ \\perp MN$. 若$MN=NQ=1$, 二面角$P-NQ-M$的大小为$\\dfrac{\\pi}{4}$, 若点$E$为棱$PN$的中点, 点$F$为棱$PQ$的中点, 那么三棱锥$M-EFN$的体积是多少?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$M$} coordinate (M);\n\\draw (1,0,1) node [below] {$N$} coordinate (N);\n\\draw (2,0,0) node [right] {$Q$} coordinate (Q);\n\\draw (0,{sqrt(2)},0) node [above] {$P$} coordinate (P);\n\\draw (P)--(M)--(N)--(Q)--cycle(P)--(N);\n\\draw [dashed] (M)--(Q);\n\\draw ($(P)!0.5!(Q)$) node [above right] {$F$} coordinate (F);\n\\draw ($(P)!0.5!(N)$) node [above] {$E$} coordinate (E);\n\\draw (M)--(E)--(F)--(N);\n\\draw [dashed] (M)--(F);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018723": {
+        "id": "018723",
+        "content": "如图, 已知正方体$ABCD-A_1B_1C_1D_1$的棱长为$2$, 长为$2$的线段$MN$的一个端点$M$在棱$DD_1$上运动, 点$N$在正方体的底面$ABCD$内运动, 则$MN$的中点$P$的轨迹与正方体从顶点$D$出发的三个面所围成的几何体的表面积是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A_1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B_1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C_1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D_1);\n\\draw (A_1) -- (B_1) -- (C_1) -- (D_1) -- cycle;\n\\draw (A) -- (A_1) (B) -- (B_1) (C) -- (C_1);\n\\draw [dashed] (D) -- (D_1);\n\\filldraw ({sqrt(3)/2},0,{-3/2}) circle (0.03) node [right] {$N$} coordinate (N);\n\\filldraw (0,1,-2) circle (0.03) node [left] {$M$} coordinate (M);\n\\filldraw ($(M)!0.5!(N)$) circle (0.03) node [right] {$P$} coordinate (P);\n\\draw [dashed] (M)--(N);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018724": {
+        "id": "018724",
+        "content": "如图, 三棱锥$P-MNQ$中, $PM \\perp NQ$, $PM \\perp MN$, $NQ \\perp MN$, 若$MN=1$, $PQ=2$, 求三棱锥$P-MNQ$体积的最大值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$M$} coordinate (M);\n\\draw ({2/sqrt(10)},0,{sqrt(6/10)}) node [below] {$N$} coordinate (N);\n\\draw ({sqrt(10)/2},0,0) node [right] {$Q$} coordinate (Q);\n\\draw (0,{sqrt(6)/2},0) node [above] {$P$} coordinate (P);\n\\draw (P)--(M)--(N)--(Q)--cycle(P)--(N);\n\\draw [dashed] (M)--(Q);\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [
+            "第六单元"
+        ],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第三册简单几何体例题与习题",
+        "edit": [
+            "20230706\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
     "020001": {
         "id": "020001",
         "content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.",