isogonal conjugate

Let ABC be a triangle, AL the angle bisectorMathworldPlanetmath of BAC and AX any line passing through A. The isogonal conjugateMathworldPlanetmath line to AX is the line AY obtained by reflecting the line AX on the angle bisector AL.

In the picture YAL=LAX. This is the reason why AX and AY are called isogonal conjugates, since they form the same angle with AL. (iso= equal, gonal = angle).

Let P be a point on the plane. The lines AP,BP,CP are concurrentMathworldPlanetmath by construction. Consider now their isogonals conjugates (reflectionsPlanetmathPlanetmath on the inner angle bisectors). The isogonals conjugates will also concurr by the fundamental theorem on isogonal lines, and their intersection point Q is called the isogonal conjugate of P.

If Q is the isogonal conjugate of P, then P is the isogonal conjugate of Q so both are often referred as an isogonal conjugate pair.

An example of isogonal conjugate pair is found by looking at the centroid of the triangle and the Lemoine point.

Title isogonal conjugate
Canonical name IsogonalConjugate
Date of creation 2013-03-22 13:01:13
Last modified on 2013-03-22 13:01:13
Owner drini (3)
Last modified by drini (3)
Numerical id 7
Author drini (3)
Entry type Definition
Classification msc 51-00
Related topic SymmedianMathworldPlanetmath
Related topic LemoinePoint
Related topic FundamentalTheoremOnIsogonalLines
Defines isogonal conjugate pair
Defines isogonal