Euclidea
趣味几何游戏。Website
热知识：尺规做图中的直尺无刻度，不可用于测量。
冷知识：使用单圆规可以达到与尺规做图同样的效果，即纸上无线，心中有线。
冷知识：尺规操作中的圆规为「松」圆规，即圆规离纸后两脚立即合拢，不能直接转移长度。但可以通过数学证明确定「松」圆规加直尺可以达到能够转移长度的「紧」圆规同样效果。
使用尺规操作完成对应题目。每道题目另有 L 目标及 E 目标，部分题目有隐藏 V 目标。L 目标为操作步骤不大于要求数。E 目标为操作花费不大于要求数。V 目标为找到符合题目要求的全部解答。不同的目标是独立的，不需要在一次操作内全部达成。
可执行的操作及对应花费如下。
| Action | Cost |
|---|---|
| line | 1 |
| cycle | 1 |
| Perpendicular Bisector | 3 |
| Perpendicular | 3 |
| Angle Bisector | 4 |
| Parallel | 4 |
| Compass | 5 |
[TOC]
1.1 Angle of 60॰
Construct an angle of 60॰ with the given side.
给定一条射线，以此为边作 60 度角。
3L 3E 2V
1.2 Perpendicular Bisector
Construct the perpendicular bisector of the segment.
作给定线段的中垂线。（完成后获得中垂线工具。）
3L 3E
1.3 Mid point
Construct the midpoint of the segment defined by two points.
找到给定两点的中点。
2L 4E
1.4 Cycle in Square
Inscribe a circle in the square.
作给定正方形内接圆。
3L 5E
1.5 Rhombus in Rectangle
Inscribe a rhombus in the rectangle so that they share a diagonal.
作给定长方形内接菱形（一组对边在矩形对边上，一对顶点与矩形顶点重合）。
3L 5E 2V
1.6 Cycle Center
Construct the center of the circle.
找到给定圆的圆心。
2L 5E
E解：在给定圆上作 3 个半径相等的圆，得到两条弦的中垂线，交点即为圆心。
1.7 Inscribed Square
Inscribe a square in the circle. One vertex of the square is given.
做给定圆心的圆内接正方形.
E解: 圆上一点 A, 以 OA 为半径作圆 A 交圆 O 于 B, C. 以 BC 为半径作圆 C 交圆 A 于点 D, 交圆 O 于点 E. 直线 OC 交圆 C 于点 F, G. 直线 EF 交圆 O 于 H. 直线 EG 交圆 O 于点 I.
A, H, E, I 为圆内接正方形的四顶点.
证明：以 O 为原点建立座标系，设 A 的座标为 (0, 1)。则:
C $(\frac {\sqrt 3} 2, \frac 1 2)$。
CF=CB=$\sqrt 3$
F $(\frac {\sqrt 3+3} 2, \frac {1+\sqrt 3} 2)$
G $(-\frac {3-\sqrt 3} 2, -\frac {\sqrt 3-1} 2)$
直线 EF 的方程为：$y=x-1$，其与圆 O 的交点恰好为 H (1, 0)
直线 EG 的方程为：$y=-x-1$，I (-1, 0).
2.1 Angle Bisector
Construct the line that bisects the given angle.
作角平分线。（完成后获得角平分线工具）
2L 4E
2.2 Intersection of Angle Bisectors
Construct the point where the angle bisectors of the triangle are intersected.
找到三角形内接圆圆心。
2L 6E
E 解：以三角形最短边为半径作 4 个圆，得到两条角平分线，交点即为内心。
2.3 Angle of 30॰
Construct an angle of 30॰ with the given side.
给定一条射线，以此为边作 30 度角。
3L 3E 2V
在射线上连续作两个圆，连接端点与交点即得到 30 度角。
2.4 Double Angle
Construct an angle equal to the given one so that they share one side.
将一个给定角度翻倍。
3L 3E 2V
2.5 Cut Rectangle
Construct a line through the given point that cuts the rectangle into two parts of equal area.
作一条直线经过给定点，并将给定矩形切成全等两部分。
3L 3E
2.6 Drop a Perpendicular
Drop a perpendicular from the point to the line.
过直线外一点作直线的垂线。
2L 3E
以直线上任意选取的两个点为圆心，到给定点为半径作两个圆，连接两圆交点即为直线的垂线。
2.7 Erect a Perpendicular
Erect a perpendicular from the point on the line.
过直线上一点作直线的垂线.（完成后获得垂线工具。）
1L 3E
E 解: 以直线外一点作圆 O 交直线于 A, B. 直线 AO 交圆 O 于点 C. 则 BC 为直线的垂线.
2.8 Tangent to Circle at Point
Construct a tangent to the circle at the given point.
作给出圆心的圆的切线.
2L 3E
E 解: 取圆上一点 A, 以小于圆 O 直径的某一距离为半径作圆 A 交圆 O 于 B, C. 以 BC 为半径作圆 C 交圆 A 于 D.
CD 即为圆 O 切线.
2.9 Circle Tangent to Line
Construct a circle with the given center that is tangent to the given line.
作与直线相切的圆。
2L 4E
2.10 Cycle in Rhombus
Inscribe a circle in the rhombus.
作菱形内接圆。
4L 6E
3.1 Chord Midpoint
Construct a chord whose midpoint is given.
给定一个圆及其圆心，另给圆内一点，找到以此点为中点的弦。
2L 4E
3.2 Triangle by Angle and Orthocenter
Construct a segment connecting the sides of the angle to get a triangle whose orthocenter is in the point O.
给定一个角及角内一点，找到三角形的第三条边使该点为三角形垂心。
3L 6E
3.3 Intersection of Perpendicular Bisectors
Construct a segment connecting the sides of the angle to get a triangle whose perpendicular bisectors are intersected in the point O.
给定一个角及角内一点，找到三角形的第三条边使该点为三角形外心。
2L 2E
3.4 Three equal segments - 1
Given an angle ABC and a point M inside it, find points D on BA and E on BC and construct segments DM and ME such that BD = DM = ME.
给定一个角及角内一点，找到分别位于两边上的两点使三段距离相等。
4L 6E 2V
3.5 Circle through Point Tangent to Line
Construct a circle through the point A that is tangent to the given line at the point B.
给定一条直线及直线上、直线外各一点，作与直线相切并经过两点的圆。
3L 6E
3.6 Midpoints of Trapezoid Bases
Construct a line passing through the midpoints of the trapezoid bases.
作一直线经过给定梯形上下底中点。
3L 5E
3.7 Angle of 45॰
Construct an angle of 45॰ with the given side.
作 45 度角。
2L 5E 2V
作两个半径相等的圆 A、B，交于 C、D。以 CD 为半径作圆 C，与直线 AC 相交于 E。角 AOE 为 45 度。
证明：
设 A 点座标为 (1, 0)
则 C 点座标为 $ \left( \frac{3}{2}, \frac{\sqrt3}{2}\right) $
CE 长度为 $ \sqrt3 $
E 点座标为 $ \left( \frac{3}{2} + \frac{\sqrt{3}}{2} , \frac{3}{2} + \frac{\sqrt{3}}{2}\right) $
即直线 OE 斜率为 1，角 AOE 为 45 度。
3.8 Lozenge
Construct a rhombus with the given side and an angle of 45॰ in a vertex.
给定一条线段，作以此为边的 45 度角菱形。
5L 7E 4V
4.1 Double Segment
| Construct a point C on the line AB such that | AC | = 2 | AB | using only a compass. |
| 使用单规做图，找到直线 AB 上的点 C 满足 | AC | =2* | AB | 。 |
3L 3E 2V
4.2 Angle of 60deg -2
Construct a straight line through the given point that makes an angle of 60॰ with the given line
过直线外一点作一条与给定直线成 60 度角的直线。
3L 4E 2V
4.3 Circumscribed Equilateral Triangle
Construct an equilateral triangle that is circumscribed about the circle and contains the given point.
作给定圆的外接正三角形，其中一条边经过圆上给定一点。
5L 6E
4.4 Equilateral Triangle in Circle
Inscribe an equilateral triangle in the circle using the given point as a vertex. The center of the circle is not given.
作圆内接正三角形。
5L 6E
4.5 Cut Two Rectangles
Construct a line that cuts each of the rectangles into two parts of equal area.
作一条直线分割两个矩形，使每个矩形分割成的两部分全等。
5L 5E
4.6 Square Root of 2
Let |AB| = 1. Construct a point C on the ray AB such that the length of AC is equal to $ \sqrt 2 $.
3L 5E
4.7 Square Root of 3
Let |AB| = 1. Construct a point C on the ray AB such that the length of AC is equal to $ \sqrt 3 $.
3L 3E
4.8 Angle of 15॰
Construct an angle of 15॰ with the given side.
3L 5E 2V
4.9 Square by Opposite Midpoints
Construct a square, given two midpoints of opposite sides.
6L 10E
4.10 Square by adjacent Midppoints
Construct a square, given two midpoints of adjacent sides.
7L 10E 2V
5.1 Parallel Line
Construct a line parallel to the given line through the given point.
给定一条直线和直线外一点，作经过此点的直线平行线。
2L 4E
E 解：以给定点为圆心作圆，再在直线上作同半径的两个圆，第一个圆与第三个圆的交点在平行线上。
5.2 Parallelogram by Three Vertices
Construct a parallelogram whose three of four vertices are given.
给定三个不在同一直线上的点，作以此三点为顶点的平行四边形。
4L 8E 3V
E 解：用上题的办法作平行线，但此题目可以少画两个圆。
5.3 Line Equidistant from Two Points - 1
Construct a line through the point C and at equal distance from the points A and B but that does not pass between them.
2L 4E
5.4 Line Equidistant from Two Points - 2
Construct a line through the point C that goes between the points A and B and that is at equal distance from them.
3L 5E
5.5 Hash
Construct a line through the given point on which two pairs of parallel lines cut off equal line segments.
2L 4E 2V
5.6 Shift Angle
Construct an angle from the given point that is equal to the given angle so that their sides are parallel.
2L 6E
5.7 Line Equidistant from Two Lines
Construct a straight line parallel to the given parallel lines that lies at equal distance from them.
2L 5E
5.8 Circumscribed Square
Circumscribe a square about the cicle. Two of its sides should be parallel to the given line.
6L 11E
5.9 Square in Square
Inscribe a square in the square. A vertex is given.
6L 7E
5.10 Circle Tangent to Square Side
Construct a circle that is tangent to a side of the square and goes through the vertices of the opposite side.
3L 6E 4V
5.11 Regular Hexagon
Construct a regular hexagon with the given side.
6.1 Point Reflection
Reflect the segment across the point.
6.2 Reflection
Reflect the segment across the line.
5L 5E
6.3 Copy Segment
Construct a segment from the given point that is equal to the given segment and lies on the same line with it.
3L 4E
6.4 given Angle bisector
Construct two straight lines through the two given points respectively so that the given line is a bisector of the angle that they make.
4L 4E
6.5 Non-collapsing Compass
Construct a circle with the given center and the radius eual to the length of the given segment.
4L 5E
6.6 Translate Segment
Construct a segment from the given point parallel and equal to the given segment.
2L 6E 2V
6.7 Triangle by Three sides
Construct a triangle with the side AB and the two other sides equal to the given segments.
4L 12E 4V
6.8 Parallelogram
Construct a parallelogram with the given side and the midpoint of the opposite side in the given point.
5L 8E
6.9 Nine Point Circle
Construct a circle that passes through the midpoints of sides of the given acute triangle.
5L 9E