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Square Circles (Posted on 2004-05-27) Difficulty: 3 of 5

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.


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?

See The Solution Submitted by Axorion    
Rating: 4.0000 (3 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Fewer than I thought | Comment 17 of 24 |

My first thought was 3 solutions, then I realized two of them are the same.  I figured I would post before looking at all the comments.

C can form either the right angle or one of the others.

Right angle not at C:  (20 + c)^2 + (30 + c)^2 = 50^2

yields the positve solution c=10 for a (30,40,50) right triangle.

Right angle at C: 50^2 + (20 + x)^2 = (30 + x)^2

yields the positive solution c=100 for a (50,120,130) right triangle.

These should be the only solutions unless I'm missing something obvious.


  Posted by Jer on 2004-05-28 09:44:39
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