Prove that the sides of the orthic triangle meet the sides of the given triangle in three collinear points.

euclidean-geometrygeometrytriangles

Prove that the sides of the orthic triangle($DEF$) meet the sides of the given triangle($ABC$) in three collinear points($X, Y, Z$) .
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Prove using Menelaus theorem, Stewart's theorem, ceva's theorem

I tried to find the ratios $\frac{YA}{YB}$, $\frac{XA}{XC}$ and $\frac{ZB}{ZC}$ by getting individual values of these lengths via calculating some lengths by pythagoras and some by stewarts theorem. But I ended up with some complicated equations which when I tried to solve I was getting stuck in variables.

Best Answer

As you have mentioned menelaus, i would like to post the proof using cross ratios without defining cross ratios (means a little,easy exercise for you). I will avoid directed lengths.

It is well known that $ (XE,AC)=-1$ or $\frac{CE}{CX}\frac{AX}{AE}=1$ (show this!). Similiarly, $(BA,FY)=-1$ or $\frac{YA}{YB}\frac{FB}{FA}=1$ (show this!) and finally, $(CB,DZ)=-1$ or $\frac{ZB}{ZC}\frac{DC}{DB}=1$ (show this!). Now multiply all these to get the resukt immediately.

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