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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:
[1] Bečvář, J., Dlab, V.: Bohatství pýthagorejských tvrzení včetně Pythagorovy věty pro čtyři i více bodů v prostoru. Pokroky matematiky, fyziky a astronomie, 62 (2017), 4, 283–294.
[2] Dlab, V., Williams, K. S.: The many sides of the Pythagorean theorem. The College Mathematics Journal, 50 (2019), 3, 162–172. DOI 10.1080/07468342.2019.1580527 | MR 3955325
[3] Kuřina, F.: 10 pohledů na geometrii. ALBRA, Praha, 1996.
[4] Maor, E.: The Pythagorean Theorem. Princeton University Press, Princeton, 2007. MR 2316578
[5] van der Waerden, B.: Science awakening. Oxford University Press, New York, 1961. MR 0124984
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