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).