Circles and Right Triangles

Geometry Level 2

I draw 3 externally tangent circles with integer radii 1, 2, and 3, respectively.

As shown in the diagram, when I connect their centers, it forms a right triangle with integer side lengths 3-4-5.

Is it always possible to form any right triangle with integer side lengths using 3 externally tangent circles with integer radii?

Yes No

1 solution

Tom Engelsman
Mar 24, 2018

Let us take the three radii as $r_1, r_2, r_3 \in \mathbb{N}$ such that the sum of any two forms the sides & hypotenuse of a right triangle. This implies:

$r_1 + r_2 = m^2 - n^2$ ;

$r_1 + r_3 = 2mn$ ;

$r_2 + r_3 = m^2 + n^2$

for $m, n \in \mathbb{N}$ and $m > n$ . Solving this 3x3 system produces:

$\boxed{r_1 = mn - n^2; r_2 = m^2 - mn; r_3 = n^2 + mn}$

which retains our original condition of $r_1, r_2, r_3 \in \mathbb{N}$ ! So the answer is $\boxed{YES}.$

Circles and Right Triangles

1 solution

0 pending reports