fundamental theorem on isogonal lines

Let $\mathrm{△}ABC$ be a triangle and $AX,BY,CZ$ three concurrent lines at $P$. If $A{X}^{\prime },B{Y}^{\prime },C{Z}^{\prime }$ are the respective isogonal conjugate lines for $AX,BY,CZ$, then $A{X}^{\prime },B{Y}^{\prime },C{Z}^{\prime }$ 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
Canonical name	FundamentalTheoremOnIsogonalLines
Date of creation	2013-03-22 13:01:16
Last modified on	2013-03-22 13:01:16
Owner	drini (3)
Last modified by	drini (3)
Numerical id	4
Author	drini (3)
Entry type	Theorem
Classification	msc 51-00
Related topic	Isogonal
Related topic	IsogonalConjugate
Related topic	LemoinePoint
Related topic	Symmedian
Related topic	Triangle