# Difference between revisions of "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. ! Missing $inserted.)\triangle ABC$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)