MathDB
point on line connecting PQ, projections of H on AB,AC whereas AH altitude

Source: Vietnamese MO (VMO) 1974

August 20, 2018
ratioprojectionsgeometryequal anglesconstant

Problem Statement

Let ABCABC be a triangle with A=90o,AHA = 90^o, AH the altitude, P,QP,Q the feet of the perpendiculars from HH to AB,ACAB,AC respectively. Let MM be a variable point on the line PQPQ. The line through MM perpendicular to MHMH meets the lines AB,ACAB,AC at R,SR, S respectively. i) Prove that circumcircle of ARSARS always passes the fixed point HH. ii) Let M1M_1 be another position of MM with corresponding points R1,S1R_1, S_1. Prove that the ratio RR1/SS1RR_1/SS_1 is constant. iii) The point KK is symmetric to HH with respect to MM. The line through KK perpendicular to the line PQPQ meets the line RSRS at DD. Prove that∠BHR=∠DHR,∠DHS=∠CHS \angle BHR = \angle DHR, \angle DHS = \angle CHS.
Back to ProblemsView on AoPS