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Menelaus' Theorem

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Menelaus' Theorem deals with the collinearity of points on each of the three sides (extended when necessary) of a triangle. It is named for Menelaus of Alexandria.

Statement

A necessary and sufficient condition for points P, Q, R on the respective sides BC, CA, AB (or their extensions) of a triangle ABC to be collinear is that

BP\cdot CQ\cdot AR = -PC\cdot QA\cdot RB

where all segments in the formula are directed segments.

defaultpen(fontsize(8));pair A=(7,6), B=(0,0), C=(10,0), P=(4,0), Q=(6,8), R;draw((0,0)--(10,0)--(7,6)--(0,0),blue+0.75);draw...

Proof

Draw a line parallel to QP through A to intersect BC at K:

defaultpen(fontsize(8));pair A=(7,6), B=(0,0), C=(10,0), P=(4,0), Q=(6,8), R, K=(5.5,0);draw((0,0)--(10,0)--(7,6)--(0,0),blue...

\triangle RBP \sim \triangle ABK \implies \frac{AR}{RB}=\frac{KP}{PB}

\triangle QCP \sim \triangle ACK \implies \frac{QC}{QA}=\frac{PC}{PK}

Multiplying the two equalities together to eliminate the PK factor, we get:

\frac{AR}{RB}\cdot\frac{QC}{QA}=-\frac{PC}{PB}\implies \frac{AR}{RB}\cdot\frac{QC}{QA}\cdot\frac{PB}{PC}=-1

See also

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