# 2021 JMPSC Invitationals Problems/Problem 12

## Contents

## Problem

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

## See also

- Other 2021 JMPSC Invitationals Problems
- 2021 JMPSC Invitationals Answer Key
- All JMPSC Problems and Solutions

The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition.