2021 JMPSC Invitationals Problems/Problem 12
Rectangle is drawn such that and . is a square that contains vertex in its interior. Find .
Solution 1 (Clever Construction)
We draw a line from to point on such that . We then draw a line from to point on such that . Finally, we extend to point on such that .
Next, if we mark as , we know that , and . We repeat this, finding , so by AAS congruence, . This means , and , so . We see , while . Thus, ~Bradygho
Solution 2 (Trig)
Let . We have , and . Now, Law Of cosines on and gets and , so ~ Geometry285
Solution 3 (Mass points and Ptolemy)
Let be the center of square . Applying moment of inertia to the system of mass points (which has center of mass ) gives Since is a right triangle, we may further cancel out some terms via Pythag to get To compute , apply Ptolemy to cyclic quadrilateral (using the fact that is 45-45-90) to get . Thus ~djmathman
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