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# Geometry question

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6)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.

Picture: https://latex.artofproblemsolving.com/c/5/0/c50a548001dc09c6eb63031398f6a77787276a50.png

Thanks

Mar 2, 2020

#1
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By power of a point, TU = 6*sqrt(3).

Mar 3, 2020
#2
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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.

First  case:  both radii are identical:  r = 7.5

TU = 12

Second  case:  radii  are different:    r1 = 5     r2 =  13.094

The result is the same:                      TU = 12  Mar 3, 2020
edited by Dragan  Mar 3, 2020
edited by Dragan  Mar 4, 2020
edited by Dragan  Mar 5, 2020
edited by Dragan  Mar 5, 2020
#3
+1

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.

https://latex.artofproblemsolving.com/c/5/0/c50a548001dc09c6eb63031398f6a77787276a50.png Mar 3, 2020
edited by heureka  Mar 3, 2020