Ruler and Compass Constructions

by Ken Brakke

Illustrated by JavaSketchpad

Clicking on the number link will display the construction; be patient the first time since Java may take a few moments to load (and you may have to hit Reload a time or two). In the construction, you may drag the original problem-defining points around to see the construction diagram morph.

You can follow the steps of the construction by clicking on the buttons at the left.

Basic constructions:

1. Perpendicular bisector of given segment.
2. Line perpendicular to given line through given point not on given line.
3. Right angle at given point on given line.
4. Square with given segment as side.
5. Equilateral triangle with given segment as side.
6. Hexagon with given segment as side.
7. Copy a given angle to a given segment.
8. Line parallel to given line through point not on given line.
9. Dividing given segment into N equal parts.
10. Bisecting a given angle.
11. Construct 30 degree angle on given segment.
12. Find the center of the circle through three given points.
13. Find the circumscribed circle of a given triangle.
14. Find the inscribed circle of a given triangle.
15. Construct a rectangle with two given side lengths.
16. Construct a triangle similar to a given one on a given segment.
17. Given point P on segment QR, find point C that divides given segment AB in the same ratio.
18. Construct the medians and altitudes of a given triangle.
19. Construct a golden rectangle.
20. Construct a square that has twice the area of a given square.
21. Construct a circle that has twice the area of a given circle.
22. Construct a line parallel to a given line and a given distance from it.
23. Construct a circle of a given radius tangent to two lines through a point.
24. Construct a square with the same area as a given rectangle.
25. Given a point on one side of a line, find its mirror image wrt the line.
26. Given two points on one side of a line, find the path of the ray of light between the points that reflects off the line.
27. Given a point outside a circle, find the two lines through the point tangent to the circle.
28. Given two circles, construct a circle of given radius tangent to the two circles.
29. Construct a line halfway between two given parallel lines.

Some more, if you like a challenge. These may use compound Sketchpad steps rather than individual ruler-and-compass steps.
Many of the tangent problems below use the inversion of a point in a circle, i.e. a point is mapped to a point on the same ray from the origin, but with inverse radius (taking the circle radius as unit). Inversion has the useful properties that circles map to circles, and circles through the origin map to straight lines. The inversion construction itself may be seen here. This particular construction was chosen to work regardless of whether the point is inside or outside the circle.
30. Given four arbitrary points, construct a square each of whose extended sides pass through one of the given points.
31A. Given two circles, construct the outer lines tangent to the circles.
31B. Given two circles, construct the inner lines tangent to the circles.
32. Given two parallel lines and a circle, construct a circle tangent to all three.
33. Given an angle and a circle whose center is on the angle bisector, construct the circles tangent to the sides of the angle and the circle.
34. Given and angle and a point in the interior of the angle, construct a point tangent to the sides of the angle and through the point.
35. Given an angle and an arbitrary circle, construct the circles tangent to the circle and the two sides of the angle.
36. Given a line and two points on one side of the line, construct the circle through the points and tangent to the line.
37. Given a line, circle, and point, construct a circle through the point and tangent to the line and circle.
38. Given a line and two circles, construct the circles tangent to all three.
39. Given three arbitrary circles, construct the circles tangent to all three. (Apollonius' Problem)
40. Construct a regular pentagon. (Gauss-Wantzel Theorem: a regular N-gon is constructible iff the factors of N are distinct primes among 2,3,5,17,257,65537,...)
41. Construct a regular 17-gon.
42. Construct a regular 257-gon and a regular 65537-gon.
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