# 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. Hence, construct a parallel line to line through point not on .

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 and unit segment .

9. Construct and . Hence, construct given unit segment .

10. Construct a tangent to a circle.

11. Construct common tangents to two circles.

12. Construct a parallelogram with side lengths a and b. Hence, construct a square with side length a.

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. In other words, construct the unique point on ray such that equals the square of the radius of C. Hence or otherwise, construct the inverse of a point P *using compasses only*.

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.

21. Construct given unit segment and acute angle .

22. Construct a right triangle with the given lengths of a hypotenuse and altitude to the hypotenuse.

23. Construct angles. Hence or otherwise, construct a right triangle whose median to the hypotenuse is equal to the geometric mean of the legs. *(Source: IMO)*

Good luck!