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P21 [Geometry] - Turkish NMO 1st Round - 2003

Source:

May 16, 2014
geometry

Problem Statement

The circle C1C_1 and C2C_2 are externally tangent to each other at TT. A line passing through TT meets C1C_1 at AA and meets C2C_2 at BB. The line which is tangent to C1C_1 at AA meets C2C_2 at DD and EE. If D∈[AE]D \in [AE], ∣TA∣=a|TA|=a, ∣TB∣=b|TB|=b, what is ∣BE∣|BE|?
<spanclass=′latex−bold′>(A)</span> a(a+b)<spanclass=′latex−bold′>(B)</span> a2+b2+ab<spanclass=′latex−bold′>(C)</span> a2+b2−ab<spanclass=′latex−bold′>(D)</span> a2+b2<spanclass=′latex−bold′>(E)</span> (a+b)b <span class='latex-bold'>(A)</span>\ \sqrt{a(a+b)} \qquad<span class='latex-bold'>(B)</span>\ \sqrt{a^2+b^2+ab} \qquad<span class='latex-bold'>(C)</span>\ \sqrt{a^2+b^2-ab} \qquad<span class='latex-bold'>(D)</span>\ \sqrt{a^2+b^2} \qquad<span class='latex-bold'>(E)</span>\ \sqrt{(a+b)b}
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