Gergonne's Theorem

Theorem (Gergonne, 1818)

Let $\triangle ABC$ be a triangle and points $M$ , $N$ , and $P$ to be points on sides $BC, AC,$ and $AB$ respectively such that lines $AM, BN,$ and $CP$ are concurrent at point $O.$ Then, $\frac{OM}{AM} + \frac{ON}{BN} + \frac{OP}{CP} = 1$ . $[asy] size(200); pair A,B,C,M,N,P,O; A=(0,0); B=(4,0); C=(1,3); M=(B+C)/2; N=(A+C)/2; P=(A+B)/2; O=(1.5,1); draw(A--B--C--cycle); draw(A--M); draw(B--N); draw(C--P); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$M$",M,E); label("$N$",N,W); label("$P$",P,S); label("$O$",O,NE); [/asy]$

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Gergonne's Theorem

Theorem (Gergonne, 1818)