# 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).

Title fundamental theorem on isogonal lines FundamentalTheoremOnIsogonalLines 2013-03-22 13:01:16 2013-03-22 13:01:16 drini (3) drini (3) 4 drini (3) Theorem msc 51-00 Isogonal IsogonalConjugate LemoinePoint Symmedian Triangle