2019 AIME II Problems/Problem 11
Triangle has side lengths and Circle passes through and is tangent to line at Circle passes through and is tangent to line at Let be the intersection of circles and not equal to Then where and are relatively prime positive integers. Find
-Diagram by Brendanb4321
Note that from the tangency condition that the supplement of with respects to lines and are equal to and , respectively, so from tangent-chord, Also note that , so . Using similarity ratios, we can easily find However, since and , we can use similarity ratios to get Now we use Law of Cosines on : From reverse Law of Cosines, . This gives us so our answer is . -franchester
On the Spot STEM solves this problem here: https://www.youtube.com/watch?v=nJydO5CLuuI
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