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In right triangle $$ABC,$$  $$AC=BC$$ and $$\angle C = 90^\circ.$$ Let $$P$$ and $$Q$$ be points on hypotenuse $$\overline{AB},$$ as shown below, such that $$\angle PCQ = 45^\circ.$$ Show that $$AP^2 + BQ^2 = PQ^2.$$

Thank you very much!

May 8, 2020

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here's a hint: reflect the triangle around C so that AC is on BC and look for similar triangles

i'll write out a whole solution in a bit but that's mainly what you need to figure it out

May 8, 2020
edited by chrissy  May 8, 2020