Let ABC be a triangle with A=90o,AH the altitude, P,Q the feet of the perpendiculars from H to AB,AC respectively. Let M be a variable point on the line PQ. The line through M perpendicular to MH meets the lines AB,AC at R,S respectively.
i) Prove that circumcircle of ARS always passes the fixed point H.
ii) Let M1 be another position of M with corresponding points R1,S1. Prove that the ratio RR1/SS1 is constant.
iii) The point K is symmetric to H with respect to M. The line through K perpendicular to the line PQ meets the line RS at D. Prove that∠BHR=∠DHR,∠DHS=∠CHS. ratioprojectionsgeometryequal anglesconstant