Let M1, M2, and M3 be the points of tangency of
O1(r1), O2(r2), and O3(r3) with chord AB.
Let N1 and N3 be the projections of O1 and O3on CD.
|M1M3| = |M1M2| + |M2M3|
= 2√(r2)√(r1) + 2√(r2)√(r3) (1)
|M1M3| = |M1M| + |MM3|
= √(|O1O|2 - |ON1|2) + √(|O3O|2 - |ON3|2)
= √([r - r1]2 - [r - (2r2 + r1]2) + √([r - r3]2 - [r - (2r2 + r3]2)
= 2√r2√(r - r2 - r1) + 2√r2√(r - r2 - r3) (2)
Combining (1) and (2) gives
√(r - r2 - r1) - √(r1) = √(r3) - √(r - r2 - r3)
If r1 = r3, then
√(r - r2 - r1) = √(r1)
or
r = r1 + r2 + r1 = r1 + r2 + r3
If r1 ≠ r3, then squaring both sides gives
√(r1)√(r - r2 - r1) = √(r3)√(r - r2 - r3)
Squaring both sides again gives
(r1 - r3)r = (r1 - r3)r2 + (r1 - r3)(r1 + r3)
or
r = r1 + r2 + r3
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