Question

Prove the following:

The sides of the orthic triangle meet the sidesof the given triangle in three collinear points.

Answer 1

In the above drawing, where AD, BE, CF are the three heights of triangle ABC and lines EF, DE, and FD cut sides BC, AB and CA extended at X, Y and Z, we readily see from the complete quadrangle AFHE that D and X are harmonic conjugates with respect to B and C; therefore XB:XC = - DB:DC. Likewise, YA:YB = -FA:FB and ZC:ZA = -EC:EA. It follows that YA:YB * XB:XC * ZC:ZA = -FA:FB * DB:DC *EC:EA. But since AD, BE, CZ are concurrrent, by Ceva’s theorem the second product of ratios is -1; it follows that the first product of ratios =+1, and therefore, by the converse of Menelaus’ theorem, that X, Y, Z are collinear.

In fact, this reasoning proves something more general: if D, E, F are points on the sides BC, CA and AB of a triangle such that AD, BE and CF are concurrent, then lines EF, DE, and FD cut sides BC, AB and CA at collinear points.