fundamental theorem on isogonal lines

Let ABC be a triangleMathworldPlanetmath and AX,BY,CZ three concurrent lines at P. If AX,BY,CZ are the respective isogonal conjugateMathworldPlanetmath lines for AX,BY,CZ, then AX,BY,CZ are also concurrentMathworldPlanetmath at some point P.

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

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