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.

2     Possible solutions

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

2.1    Hexagon, square, and triangle

Fig. 1: 6, 4, 3

The right triangle is half a rectangle in the DIN format (European paper format).

2.2    Two hexagons and a square

Fig. 2: 6, 6, 4

The right triangle is half a square

2.3    Decagon, hexagon, and pentagon

Fig. 3: 10, 6, 5

The right triangle is half a golden rectangle.

2.4    Decagon, decagonal star, and triangle

Fig. 4: 10, 10/3, 3

The right triangle is half a long golden rectangle with sides  and .

2.5    Hexagon, decagonal star, and pentagram

Fig. 5: 6, 10/3, 5/2

The right triangle is half a golden rectangle.

3     Two-gons

If we allow regular two-gons, i. e. diameters, we get infinitely many solutions. Table 2 gives the first solutions. The two-gon 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: Two-gons included

Fig. 6: 6, 3, 2

3.2    14/3, 7/2, 2

Fig. 7: 14/3, 7/2, 2

3.3    General case

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.