Construction

Constructions with straight edge and compass (i.e. the ability to mark off segments, draw circles and arcs, and draw straight lines) are a branch of geometry that rely on the use of basic geometrical axioms to create various figures in the Euclidean plane.

A compass is a tool that can draw circles and arcs of circles.

A straightedge is an unmarked ruler that can draw line segments.

No other tools are allowed in a construction. However, the two basic tools alone can allow one to:

1. Duplicate a line segment.

2. Copy an angle.

3. Construct an angle bisector.

4. Construct a perpendicular bisector.

5. Construct a perpendicular from a point to a line.

6. Construct a triangle with side lengths a, b, and c.

7. Partition a line segment into different parts.

8. Construct length given lengths and .

9. Construct and .

10. Construct a tangent to a circle.

11. Construct a common tangents to two circles.

12. Construct a parallelogram with side lengths a and b.

These basic constructions should be easy to accomplish. Now, try these:

13. Construct a line passing through a point parallel to line .

14. Construct a square circumscribed on a circle.

15. Construct a regular hexagon inside a given circle.

16. Construct the inverse of a point P with respect to circle C.

17. Construct a square, all of whose vertices are on a given triangle.

18. Construct a regular pentagon.

19. Construct the radical axis of two circles.

20. Given two chords of a circle intersecting in the interior of the circle, construct another circle tangent to the chords and internally tangent to the original circle.

Good luck!