Construction

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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 $l$ through point $A$ not on $l$.

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 $n$ different parts.

8. Construct length $ab$ given lengths $a$ and $b$ and unit segment $1$.

9. Construct $a/b$ and $\sqrt{ab}$. Hence, construct $\sqrt{a}$ given unit segment $1$.

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 $P$ parallel to line $l$.

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 $P'$ on ray $CP$ such that $CP * CP'$ 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 $sin C, cos C, tan C$ given unit segment $1$ and acute angle $C$.

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

23. Construct $15^\circ, 30^\circ, 45^\circ, 60^\circ, 75^\circ$ 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!

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