Problem: (1986 China TST – #5) Given a square whose side length is , and are points on the sides and , respectively. If the perimeter of is find the angle .
Solution: Let and . By the given condition, we have (1). From this, we find
Substituting from (1), we have
Note that and . Then
But by (2), we have .
Hence . QED.
Comment: Trigonometry can come in handy quite often, especially when dealing with angles. Purely geometric solutions to this problem are a lot more complicated in my opinion; when in doubt, use algebra. The following identity comes in handy on quite a few problems.
Practice Problem: Prove that in any triangle we have .
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