Carnot's Theorem

Carnot's Theorem states that in a triangle $ABC$ with $A_1\in BC$ , $B_1\in AC$ , and $C_1\in AB$ , perpendiculars to the sides $BC,$ AC $, and$ AB $at$ A_1 $,$ B_1 $, and$ C_1 $are [[concurrent]] [[if and only if]]$ A_1B^2+C_1A^2+B_1C^2=A_1C^2+C_1B^2+B_1A^2$.

==Proof== {{incomplete|proof}}

==Problems== ===Olympiad===$ (Error compiling LaTeX. Unknown error_msg)\triangle ABC $is a triangle. Take points$ D, E, F $on the perpendicular bisectors of$ BC, CA, AB $respectively. Show that the lines through$ A, B, C $perpendicular to$ EF, FD, DE$ respectively are concurrent. (Source)

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Carnot's Theorem

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