Problem

Source: VMO 2023 day 1 P4

Tags: geometry



Given is a triangle ABC and let D be the midpoint the major arc BAC of its circumcircle. Let M,N be the midpoints of AB,AC and J,E,F are the touchpoints of the incircle (I) of ABC with BC,CA,AB. The line MN intersects JE,JF at K,H respectively; IJ intersects the circle (BIC) at G and DG intersects (BIC) at T. a) Prove that JA passes through the midpoint of HK and is perpendicular to IT. b) Let R,S respectively be the perpendicular projection of D on AB,AC. Take the points P,Q on IF,IE respectively such that KP and HQ are both perpendicular to MN. Prove that the three lines MP,NQ and RS are concurrent .