Sets of points in the triangle usually come in ones and threes, for obvious symmetry reasons. P , called O here, is the isogonal conjugate of P . I hadn't heard of such a thing before yesterday. If you take the 3 lines from the vertices A , B and C to P , then reflect them across the angle bissectors , you get a new triple of lines. They concur on the isogonal conjugate.
Every line is isogonal conjugate to a certain conic .
A line that crosses the circumcircle is isogonal to a hyperbola , a tangent to a parabola , a line that is wide of the circumcircle is conjugate to a circle. Thus the conjugate of the circumcircle is the line at infinity. The pages blocked vary each day. D. E. Joyce's The Geometry Applet . This is a wondeful applet.