Carnot's Theorem

Revision as of 10:21, 26 August 2008 by 1=2 (talk | contribs) (New page: '''Carnot's Theorem''' states that in a triangle <math>ABC</math> with <math>A_1\in BC</math>, <math>B_1\in AC</math>, and <math>C_1\in </math>AB<math>, perpendiculars to the sides...)
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Carnot's Theorem states that in a triangle $ABC$ with $A_1\in BC$, $B_1\in AC$, and $C_1\in$AB$, [[perpendicular]]s to the sides$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

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)

See also

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