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 iffif and only if $A_1B^2+C_1A^2+B_1C^2=A_1C^2+C_1B^2+B_1A^2$ .

Proof

Template:Incomplete

Problems

Olympiad

$\triangle ABC$ 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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Proof

Problems

Olympiad

See also