Geometry Puzzle

Below, you find a nice geometry puzzle.
Send your solution , with your name and e-mail address, to
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Solution 1 by Bruce

Given are:
- a circle with radius r1 and inside
- two circles with radius r2 and r3 which contact circle 1 at points A and B
- the centers of circles 1,2 and 3.
- the intersection point P of circles 2 and 3

Prove the following theorem:
    If P is on line AB then r1 = r2 + r3 must be true


Solution 1

The problem is to prove that line APB is straight if r1 = r2 + r3

For APB to be a straight line, we need to show that the sum of angles
APD + DPE + EPB = 180

Notice that CD + AD = r1 also CE + EB = r1
But, our condition is r1 = r2 + r3
So, this means that DPEC is a parallelogram.
So, CD = r3 and CE = r2

Finally, we use the parallelogram to relate angles

Since angles of triangle ADP sum to 180deg
Then so do the angles on the line APB.