Solution of the Week #387 - Five Circles

If you start with the smallest triangle and ensure it is a Pythagorean triple, then the radii of all of the circles is bound to be rational, and so can always be scaled up by an appropriate factor to obtain whole numbers. Beginning with a larger triangle and working backwards will not necessarily ensure this.

The best solution is found by starting with a 5-12-13 Pythagorean triple. The radii of the tangent circles for this are found by the semi-perimeter less each of the triangle edges, so 10-3-2.

Using Pythagoras on the second triangle, (10+3),(3+x),(x+10), we find that x needs to be 39/7. We scale up all the numbers we have by a factor of 7 to maintain whole numbers.

On the third and final triangle we now have (70+39),(39+x),(x+70), we find that x is 4251/31. Scaling everything up by a factor of 31, the five radii are:

2170, 434, 651, 1209 and 4251, for a total of 8715.

We might have used a different triangle to begin the construction, but we would have found that the procedure fails when the ratio between the radii of the zero-th circle and the first circle (2170 and 434 in our figure) is 4 or less. So for instance the 3-4-5 triangle would not have worked.