Construction

Revision as of 13:32, 15 June 2014 by Suli (talk | contribs)

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 $n$ different parts. 8. Construct length $ab$ given lengths $a$ and $b$. 9. Construct $a/b$ and $\sqrt{ab}$. 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 $P$ 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. 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!