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 Square Circles (Posted on 2004-05-27)
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?

 Submitted by Axorion Rating: 4.0000 (3 votes) Solution: (Hide) 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

 Subject Author Date No Subject Kevin Tran 2004-05-31 19:20:18 Sorry - I'm back Axorion 2004-05-30 04:02:30 re: Crossing my fingers Charlie 2004-05-28 15:05:04 Crossing my fingers Jer 2004-05-28 14:36:14 re(3): Fewer than I thought Charlie 2004-05-28 11:40:05 re(2): Fewer than I thought Jer 2004-05-28 10:21:32 re: Fewer than I thought Danny 2004-05-28 10:09:38 Fewer than I thought Jer 2004-05-28 09:44:39 re(3): are you from Russia? SilverKnight 2004-05-27 22:22:04 re(2): are you from Russia? Cory Taylor 2004-05-27 21:02:15 :-) Danny 2004-05-27 18:13:05 No idea what those o:p's are Charlie 2004-05-27 18:08:07 re: are you from Russia? Charlie 2004-05-27 18:05:50 many similar triangles Danny 2004-05-27 18:05:24 are you from Russia? Danny 2004-05-27 17:49:35 re: the values Charlie 2004-05-27 17:44:15 re(2): solution Danny 2004-05-27 17:37:57 re: solution Danny 2004-05-27 17:34:39 I'm not from Russia... Danny 2004-05-27 16:50:29 the values Charlie 2004-05-27 16:01:05 re: solution Charlie 2004-05-27 15:43:44 solution Charlie 2004-05-27 15:26:52 re: my old friend Pythagoras Charlie 2004-05-27 14:42:47 my old friend Pythagoras Danny 2004-05-27 13:52:27

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