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# Geometry Proofing Problem

Let ABCD be a rectangle and let P be a point on it circumcircle, different from any vertex. Let X, Y, Z and W be the projections of P onto the lines AB, BC, CD, and DA, respectively. Prove that one of the points X, Y, Z and W is the orthocenter of the triangle formed by the other three.

Note by Dani Natanael
4 years ago

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Hint: Lines $$AB$$ and $$CD$$ are perpendicular if and only if $$AC^2 + BD^2 = AD^2 + BC^2$$.

This is one of my favorite ways of showing that 2 lines are perpendicular.

Using the above hint, draw a diagram, and the result is almost immediate. Staff · 4 years ago