Hans Walser, [20141209]
Regular Polygons and Right Triangles
We draw circles of equal size and inscribe them regular polygons and regular star polygons. With the sides of these figures we try to form right triangles.
A brute force approach indicates the conjecture that there are only fife solutions, two of them with stars (Tab. 1).
First polygon 
Second polygon 
Third polygon 
6 
4 
3 
6 
6 
4 
10 
6 
5 
10 
10/3 
3 
6 
10/3 
5/2 
Tab. 1: Solutions
Fig. 1: 6, 4, 3
The right triangle is half a rectangle in the DIN format (European paper format).
Fig. 2: 6, 6, 4
The right triangle is half a square
Fig. 3: 10, 6, 5
The right triangle is half a golden rectangle.
Fig. 4: 10, 10/3, 3
The right triangle is half a long golden rectangle with sides and .
Fig. 5: 6, 10/3, 5/2
The right triangle is half a golden rectangle.
If we allow regular twogons, i. e. diameters, we get infinitely many solutions. Table 2 gives the first solutions. The twogon is always the third polygon.
First polygon 
Second polygon 
Third polygon 
4 
4 
2 
6 
3 
2 
6 
4 
3 
6 
6 
4 
8 
8/3 
2 
10 
5/2 
2 
10 
6 
5 
10 
10/3 
3 
5 
10/3 
2 
6 
10/3 
5/2 
12 
12/5 
2 
14 
7/3 
2 
14/3 
7/2 
2 
7 
14/5 
2 
16 
16/7 
2 
16/3 
16/5 
2 
16 
16/7 
2 
18 
9/4 
2 
9/2 
18/5 
2 
9 
18/7 
2 
20 
20/9 
2 
20/3 
20/7 
2 



Tab. 2: Twogons included
Fig. 6: 6, 3, 2
Fig. 7: 14/3, 7/2, 2
One of the two first polygons is arbitrary. The second polygon is such that each side is orthogonal to a side of the first polygon. The circles of the two polygons are tangent.