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Find AD when area of ABC is maximal [Taiwan 2014 Quizzes]

Source:

July 18, 2014
geometrytrigonometrygeometric transformation

Problem Statement

Let O1O_1, O2O_2 be two circles with radii R1R_1 and R2R_2, and suppose the circles meet at AA and DD. Draw a line LL through DD intersecting O1O_1, O2O_2 at BB and CC. Now allow the distance between the centers as well as the choice of LL to vary. Find the length of ADAD when the area of ABCABC is maximized.
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