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Square Circles (Posted on 2004-05-27) Difficulty: 3 of 5
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 A*B=C 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

Comments: ( You must be logged in to post comments.)
  Subject Author Date
QuestionNo SubjectKevin Tran2004-05-31 19:20:18
Sorry - I'm backAxorion2004-05-30 04:02:30
re: Crossing my fingersCharlie2004-05-28 15:05:04
Crossing my fingersJer2004-05-28 14:36:14
re(3): Fewer than I thoughtCharlie2004-05-28 11:40:05
re(2): Fewer than I thoughtJer2004-05-28 10:21:32
Hints/Tipsre: Fewer than I thoughtDanny2004-05-28 10:09:38
SolutionFewer than I thoughtJer2004-05-28 09:44:39
re(3): are you from Russia?SilverKnight2004-05-27 22:22:04
re(2): are you from Russia?Cory Taylor2004-05-27 21:02:15
:-)Danny2004-05-27 18:13:05
No idea what those o:p's areCharlie2004-05-27 18:08:07
Solutionre: are you from Russia?Charlie2004-05-27 18:05:50
many similar trianglesDanny2004-05-27 18:05:24
Hints/Tipsare you from Russia?Danny2004-05-27 17:49:35
re: the valuesCharlie2004-05-27 17:44:15
Hints/Tipsre(2): solutionDanny2004-05-27 17:37:57
Hints/Tipsre: solutionDanny2004-05-27 17:34:39
I'm not from Russia...Danny2004-05-27 16:50:29
Solutionthe valuesCharlie2004-05-27 16:01:05
Solutionre: solutionCharlie2004-05-27 15:43:44
SolutionsolutionCharlie2004-05-27 15:26:52
re: my old friend PythagorasCharlie2004-05-27 14:42:47
Solutionmy old friend PythagorasDanny2004-05-27 13:52:27
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