Given:

Three circles A, B and C.

Each circle is tangent to the other two.

The radius of A is 20.

The radius of B is 30.

Questions:

How many unique values of radius C exist where the centers of A, B and C form a right triangle? (Unique: Do not count triangles which are equal through flips and rotations. You may only count dissimilar triangles and similar triagles of differing sizes.)

What are the values?

(In reply to my old friend Pythagoras by GOM)

There's more to it than that:

The distance between the centers of the radius-20 circle and the radius-30 circle could be 10, rathern than 50, if one is internally tangent to the other.

The constraint on the other two sides is not that they form integers, but rather that it is possible to use the center of a circle that is tangent to the other two as a third vertex.  The specific numbers you mention might not be possible given this condition.  But remember also that the third circle might be internally tangent to one of the original circles and externally tangent to the other, etc. in different combinations.

Posted by Charlie on 2004-05-27 14:42:47