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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. Calvin Lin Staff · 4 years ago

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@Calvin Lin Thanks you (y) Dani Natanael · 3 years, 4 months ago

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