List of constructions

The following are constructions with straightedge and compass that you should know.

1. Given a segment AB, construct an equilateral triangle whose sides have the same length as AB.

2. Given a segment AB, construct a perpendicular bisector to AB. This gives a method of bisection: it yields the midpoint.

3. Given a line l and a point P on l, construct a line perpendicular to l going through P. (This is similar to 2. Make P a midpoint)

4. Given a line l and a point P not on l, construct a line perpendicular to l going through P. (Hint: draw a circle with center P, then use 2).

5. Given a line l and a point P not on l, construct a line parallel to l going through P. (Hint: do 4, then do 3).

6. Given segments UV and XY, construct a rectangle ABCD such that AB and UV have the same length, and CD and XY have the same length. As a corollary, given a segment UV, you can construct a square ABCD such that AB and UV have the same length. (Hint: do 3 several times).

7. Given segments AB and CD, construct a segment EF which is the geometric mean. So AB:EF = EF:CD. Be able to prove that the construction works. Hint: use Thales' theorem, and the similar triangles that occur in the proof of the Pythagorean theorem.)

8. Given a segment u, construct a segment w such that w:u is the golden ratio.

9. Use number 8 to construct a regular pentagon.

10. Given a segment AB, construct a hexagon with sides of length equal to that of AB. (Hint: draw a circle with radius AB).

11. Bisect any given angle.

12. Construct angles of size 45 degrees, 36 degrees, 22.5 degrees, etc. Use these to construct a regular octagon (8-gon), decagon (10-gon), 16-gon, etc. (Can you construct a 15-gon?) (Hint: bisect. Once you have the desired central angle, draw a circle, and step through with the correct length).

13. Given rectangle ABCD, construct a square EFGH of the same area (use number 7). This is called "squaring a rectangle" or the "quadrature of a rectangle".

14. Convert any rectangle to a normalized rectangle: given a rectangle ABCD, and a segment EF, construct a rectangle HIJK such that HI has length EF and HIJK has the same area as ABCD. Give a proof that this construction works. The classical name for this procedure is "application of areas". (Hint: draw a certain rectangle, use the diagonal).

15. Given triangle ABC, construct a square EFGH of the same area. This is called "squaring a triangle" or the "quadrature of a triangle". Hint: use number 4 to find the height. Bisect the base (or the hight. Now find the geometric mean.

16. Given a polygon, construct a square of the same area. This is called "squaring a polygon" or the "quadrature of a polygon". (Hint: divide the polygon into triangles). It is impossible to square a circle with a ruler and compass. However, some curved areas can be squared. For example the lunes of Hippocrates.

17. Given the lune of Hippocrates build on a 45-45-90 triangle, show that the lune has area one half the given triangle. Conclude that you can square this lune.

18. Given a square ABCD, find a square EFGA of twice the area. (One technique is to find the geometric mean of AB and twice AB. Another is to simply draw the diagonal of the original square.) This is called "duplication of the square". The Delian problem is to duplicate the cube. This cannot be done with an unmarked ruler and compass.

19. Trisect, or n-sect any given segment AB. However, you cannot trisect angles with unmarked rulers and compass. Be able to prove that the construction works.

20. Trisect a given angle using a marked ruler and compass. (Using the technique of Archimedes. The ruler only needs two marks). Be able to prove that the construction works.

NOTE: You now can construct regular 3-gons (equilateral triangles), 4-gons (squares), 5-gons (pentagons), 6-gons (hexagons), 8-gons, 10-gons, 12-gons, 16-gons, and with some thought 15-gons. When he was 19, Gauss figured out how to construct a regular 17-gon. It turns out to be impossible to construct the regular 7-gon, 9-gon, 11-gon, 13-gon, 14-gon, 18-gon, and 19-gon. In fact, the constructibility of n-gons is related to the factorization of n and so-called Fermat primes.