1999 JBMO Problems/Problem 4
Its easy to see that , , are collinear (since angle = = ).
Applying Sine rule in triangle , we get:
Since and are cyclic quadrilaterals, anlge = anlge and
So,
So Thus, (the circumcirlcles are congruent).
From right traingles and , we have:
So
Since is the midpoint of , is perpendicular to and hence is parallel to .
So area of traiangle = area of traingle and hence is independent of position of on .
By