Menelaus Theorem (non-parallel transversal)
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- GeoGebra Materials Team
The triangle's transversal is not parallel to a side
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Ken Frank, Created with GeoGebra
Drag points H and D to show various positions of the non-parallel transversal HF.
Point H is on side CB extended, while points D and F are either both interior or both exterior to the triangle.
You can think of 'walking' the sides of triangle ABC to construct the Menelaus equation. Start at any vertex of triangle ABC, such as point B. Go from B to D, D to A, A to F, F to C, C to H, and H back to B. Write the ratios as you walk. Between the letters of any two triangle vertices comes the letter of the transversal's intersection point for that side.
Do you see that it is equally true that AB is a transversal of triangle HFC? Menelaus can be written for triangle HFC as
(FD/DH) (HB/BC) (CA/AF) = 1
The Menelaus Theorem can be proved using similar triangles, as in the next diagram.