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?

If A is internally tangent to B and the center of the small circle is at the right angle, the distance between their centers is 10, the longer leg is 20+r where r is the radius of C, and the hypotenuse is 30-r, so: 10^2+(20+r)^2 = (30-r)^2; this makes r = 4.

If the center of the large circle is the right angle, then 10^2+(30-r)^2 = (20+r)^2; thus r = 6.

If  A and B are externally tangent and both are external to C, A and B are separated by 50 units.

The center of C will then be at a right angle of the formed triangle if (20+r)^2 + (30+r)^2 = 50^2, so r = 10

The right angle will be at the center of A if (20+r)^2 + 50^2 = (30+r)^2; so r = 100

If A and B are externally tangent to each other but internal to C, then if the right angle is at the center of C then, (r-30)^2 +(r-20)^2 = 50^2; so r = 60

Have to go now... leave the last case till later.

 

 

Posted by Charlie on 2004-05-27 16:01:05