The Pentagon

Given circle O: Draw lines OA and OB so that they are perpendicular to each other. Find M, the midpoint of OA. Draw a circle at M, intersecting B. This circle should also intersect OA at C. Draw a circle at B intersecting C. Circle B crosses the original circle at two points of the pentagon. You can use them to find the other two.

Another variation. Given circle O: Draw AB, the diameter of circle O. Draw OC, a line perpendicular to AB. Find the midpoint M of OC. Draw a circle at M with diameter OC (purple). Draw a line through points B and M. BM (cyan) intersects circle M (purple) at D. Draw a circle at B with radius BD (red). This circle intersects the original circle at two points of the pentagon. You can use them to find the rest!

This is a fairly easy construction. Given circle O: Draw lines OA and OB so that they are perpendicular to each other. Find M, the midpoint of OB. Connect points A and M (purple). Bisect angle OMA (red). G is where the bisector intersects OA. Draw a line from G (cyan), so that it is perpendicular to OA. This line intersects circle O at H. A and H are two points of the pentagon. You can use them to find the rest.

Another variation. Given circle O: Find any point P on the circle. Connect OP. Find M, the midpoint of OP. Construct L (blue), a line perpendicular to OP. Find N, the midpoint of OM. Draw a circle at M (red) with radius MN. This circle intersects line L (blue) at A. Draw a line through OA. OA intersects circle O at D. Draw a circle at A with radius AM. Circle A intersects OD at B. B should be between O and A. Draw a perpendicular line from B (green). This line intersects circle O at C. C and D are two points of the pentagon.

There are many instances in which you need to construct a pentagon given only one side. Here's several variations. Given segment AB: Draw two circles (purple), one at A and one at B, each with radius AB. These two circles intersect at two points, C and D. Draw a line connecting C and D (cyan). Draw a circle at D (yellow), with radius DA. Circle D intersects CD at E and it intersects circles A and B at points F and G. Draw a line through points FE and GE (red). They intersect circles A and B at points H and I. Draw two circles at I and H (blue), each with radius HB. They intersect at J. ABHJI is a regular pentagon with the given side AB.

Another variation on given sides. Given side AB: Find C, the midpoint of AB. Draw a perpendicular line to point A. Extend segment AB. Draw a circle at A and B with radius AB (purple). Circle A intersects the perpendicular line at D. Draw a circle at C with radius CD (red). Circle C intersects AB at E. Draw another circle at A (cyan), this time with radius AE. Draw a circle at B (cyan) with the same radius. The two circles intersect each other at H. They also intersect circles A and B (purple) at points G and F. ABFHG is a regular pentagon with side AB.

Draw a line segment AB (yellow). Draw a circle at A and B, both with radius AB (yellow). Find C, the midpoint of AB. Draw a perpendicular line through C. Draw a circle at C with radius AB (purple). This circle intersects the line you just drew at point D. Find E, the midpoint of DC. Connect points DB with a line. Draw a circle at point D with radius DE (blue). This circle intersects DB at F. Draw a circle at point B with radius BF (red). This circle intersects line DC at G. It also intersects the large circle A at point I. Draw a circle at point A with radius AG (red). This circle intersects the large circle B at point H. AGBIH is a pentagon.

This next construction is an odd one. The first segment has nothing to do with the outcome of the pengtagon! First, draw segment AB and find M, the midpoint of AB. Draw a circle at M (purple) with radius AB. Draw a perpendicular line at M. This line intersects circle M (purple), at points C and D. Draw a circle at C, with radius CA (yellow). Draw a line through points CA (red). Draw a circle at A with radius AM. This circle intersects AC at E. Draw a circle at C with radius CD (blue). Draw a line perpendicular to AC. This line intersects the large circle C (blue) at points F and G. Connect points FC and GC with a line. Both lines intersect the small circle C (yellow), at points H and I. A,H,I are points of the pentagon. You can use them to find the others.

This final construction is known as a Mascheroni Construction. It's unique in that it doesn't use a straighedge! Follow the algorithm very carefully! Draw circle A (yellow). Find B, a random point on circle A. Draw a circle at B with radius BA (cyan). Circle B (cyan) intersects circle A (yellow) at C. Draw a circle at C with radius CA (cyan). Circle C (cyan) intersects circle A (yellow) at D. Draw a circle at D with radius DA (cyan). Circle D (cyan) intersects circle A (yellow) at E. Draw a circle at B with radius BD and draw a circle at E with radius EC (green). They intersect at F. Draw a circle at A with radius AF (blue). This new circle intersects circle B (cyan) at G. Draw a circle at G with radius GB (purple). Circle G (purple) intersects circle A (yellow) at H. Draw a circle at H with radius HG (red). Circle H (red) intersects circle A (yellow) at points I and J. Draw two circles, one each at I and J (orange), each with radii ID and JC. They intersect at K. Draw a circle at B, this time with radius BK (gray). This circle intersects circle A (yellow) at points L and M. B,L,M are points of the pentagon. You can use them to find the other two!