\fGiven: H is the orthocenter of 4ABC; P, Q, and R are the midpoints of AB,AC, and BC; U, V , and W are the midpoints of AH, CH, and BH. Prove thatP, Q, R, U, V , and W all lie on a common circle. (Hint: Draw the circle throughthree of the pointsyou will have to figure out which ones to chooseand thenshow that the other points lie on this circle. Use cyclic quadrilaterals.) Let us change the notations a bit, for easy solution. In triangle ABC with orthocenter H, let Am,Bm,Cm be the midpoints of the sides, D,E,F be the feet of the altitudes, and Ae,Be,Ce the midpoints of HA,HB,HC. DEF is called the orthic triangle of ABC; orthic triangle being a special kind of a pedal triangle The orthocentre of ABC is the incentre of DEF which would become apparent during the course of the proof. D, E, F is apparent. What we need to prove is that Am is the midpoint of BC and Ae is the midpoint of AH; the extension holding true for all three sides. Constructions: Join Ce to Bm & E. Proof: Quad. DHEC is cyclic as < HEC= < HDC = 90 deg. Hence, < ADE=