4

Part of 2013 Federal Competition For Advanced Students, Part 1

Problems(1)

Varying circle - constant ratio

Source: Austrian Federal Competition 2013, part 1, problem 4

A

B

C

be three points on a line (in this order). For each circle

k

through the points

B

C

D

be one point of intersection of the perpendicular bisector of

BC

with the circle

k

E

be the second point of intersection of the line

AD

k

. Show that for each circle

k

, the ratio of lengths

\overline{BE}:\overline{CE}

ratiogeometryperpendicular bisectorgeometry proposed