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# fundamental theorem on isogonal lines

Let $\triangle ABC$ be a triangle and $AX,BY,CZ$ three concurrent lines at $P$. If $AX^{{\prime}},BY^{{\prime}},CZ^{{\prime}}$ are the respective isogonal conjugate lines for $AX,BY,CZ$, then $AX^{{\prime}},BY^{{\prime}},CZ^{{\prime}}$ are also concurrent at some point $P^{{\prime}}$.

An applications of this theorem proves the existence of Lemoine point (for it is the intersection point of the symmedians):

This theorem is a direct consequence of Ceva’s theorem (trigonometric version).

Related:

Isogonal,IsogonalConjugate,LemoinePoint,Symmedian,Triangle

Type of Math Object:

Theorem

Major Section:

Reference

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## Mathematics Subject Classification

51-00*no label found*

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new correction: Closure properties on languages: DCFL not closed under reversal by babou

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