The Locus of the Intersection Point of an Angular Bisector and a Height of a Triangle

Conjecture The locus of the intersection point of an angular bisector and a height of a triangle is an algebraic curve higher than the fourth order.	Please enable Java for an interactive construction (with Cinderella).
	Drag the free elements, move the point C along the straight line L, or use the animation buttons to experience the locus of the intersection point of an angular bisector and a height of a triangle.