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# geometryyyyy

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Two circles intersect at A and B. A common external tangent is tangent to the circles at T and U as shown. Let M be the intersection of line AB and TU. If AB = 9 and BM = 3 find TU.

thanks to anyone who answers! :)

~ iamhappy

Feb 25, 2021

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Consider the power of point M with respect to each of the circles.

$$MT^2 = MU^2 = MB \cdot MA = 3 \cdot 12 = 36$$

This implies $$MT = MU = 6$$

Therefore, $$TU = MT + MU = 6 +6 = 12$$

Feb 25, 2021