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.)

There are 2 solutions where no circle lays within another. One with the A=90° and one with C=90°.

There are 2 solutions where circles A and B lay within C. One with the B=90° and one with C=90°.

There are 2 solutions where circles A and C lay within B. One with the A=90° and one with B=90°.

The answer to the first question is 6

Although you could use algebra to solve for R using A²*B²=C² while using combos of adding and subtracting radiuses, there is an easier method. After solving a few you begin to see a pattern. All the triangles are similar to ether a 3,4,5 triangle or a 5,12,13 triangle.

Where no circle lays within another and C=90°
Segment AB=50, AC=30 and BC=40
R=30-20=40-30=10

Where no circle lays within another and A=90°
Segment AB=50, AC=120 and BC=130
R=120-20=130-30=100

Where circles A and B lay within B and C=90°
Segment AB=50, AC=130 and BC=120
R=120+30=130+20=150

Where circles A and B lay within C and C=90°
Segment AB=50, AC=40 and BC=30
R=40+20=30+30=60

Where circles A and C lay within B and A=90°
Segment AB=10, AC=24 and BC=26
R=24-20=30-26=4

Where circles A and C lay within B and B=90°
Segment AB=10, AC=26 and BC=24
R=26-20=30-24=6

In summation: 4, 6, 10, 60, 100 and 150

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