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Article

MSC: 51M04, 97G40
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Summary:
This article presents the following simple property of a general triangle $ABC$: Let $D$ be an arbitrary point and $A_0$, $B_0$, $C_0$ the feet of the perpendiculars from $D$ on the (possibly extended) sides $BC$, $CA$, $AB$, respectively. Then \[ |{AC}_0|^2 + |{BA}_0|^2 + |{CB}_0|^2 = |{C_0B}|^2 + |{A_0C}|^2 + |{B_0A}|^2. \] This statement is a proper generalization of the Pythagorean theorem. Surprisingly, it does not appear in textbooks or other publications.
References:
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[5] van der Waerden, B.: Science awakening. Oxford University Press, New York, 1961. MR 0124984
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