Extensions and Implications of Simson’s Line

Extract

1. Let ABC be a triangle, H its orthocentre, P a point on the circumcircle, and D, E, F the feet of the perpendiculars from P on BC, CA, AB respectively. Then it is known that D, E, F are on a straight line which bisects PH. Through P draw lines making the angle ½π–α in the same sense with PD, PE, PF, PH to meet the sides in D′, E′, F′ and a line through H perpendicular to PH in H′. Then D′, E′, F′ are also on a line which bisects PH′. Call this line the Simson line (α) of P with respect to ABC