# Point in a Square

Inside a square ABCD, a point P is selected such that

Find ∠APB.

The following solution is by Murray Klamkin.

Rotate the square through 90° around point B. Let P' be the image of P, A' that of A, and D' of D. Connect P and P'.

In ΔBPP', ^{2} = 8.

In ΔAPP', PP'^{2} + AP^{2} = AP'^{2}

Summing up, ∠APB = ∠BPP' + ∠P'PA = 45° + 90° = 135°.

A similar approach works for a similar problem where point P is at distances 3, 4, 5 to the vertices of an equilateral triangle.

## Reference

- R. Honsberger,
*More Mathematical Morsels*, MAA, 1991, pp. 4-5.

|Contact| |Front page| |Contents| |Geometry| |Up|

Copyright © 1996-2018 Alexander Bogomolny

65277901 |