MathDB

3

Part of 2006 Iran Team Selection Test

Problems(2)

Question 3

Source: Iran TST 2006

4/17/2006
Suppose ABCABC is a triangle with MM the midpoint of BCBC. Suppose that AMAM intersects the incircle at K,LK,L. We draw parallel line from KK and LL to BCBC and name their second intersection point with incircle XX and YY. Suppose that AXAX and AYAY intersect BCBC at PP and QQ. Prove that BP=CQBP=CQ.
geometrygeometry proposed
Question 3

Source: Iran TST 2006

4/29/2006
Let l,ml,m be two parallel lines in the plane. Let PP be a fixed point between them. Let E,FE,F be variable points on l,ml,m such that the angle EPFEPF is fixed to a number like α\alpha where 0<α<π20<\alpha<\frac{\pi}2. (By angle EPFEPF we mean the directed angle) Show that there is another point (not PP) such that it sees the segment EFEF with a fixed angle too.
geometryparallelogramcircumcirclegeometric transformationreflectionconicsangle bisector