Given a circle and two points on that circle, P and Q, draw the chord PQ, and label its midpoint M.

Now draw two other chords of the circle AB and CD that both pass through M.

Further, draw chords AD and BC.

Label the intersection of AD and PQ, point X.
Label the intersection of BC and PQ, point Y.
_____________________________

Prove that M is the midpoint of line segment XY.

This is what I can make out of the diagram initially:

angle BAD = angle BCD
angle ABC = angle ADC
angle BMC = angle AMD
angle AMC = angle BMD
triangle AMD is similar to triangle BMC
AD/BC = DM/MB = AM/MC
BM*MA = CM*MD = PM*MQ
PM = MQ
CY*YB = PY*YQ
AX*XD = PX*XQ

Edited on March 25, 2004, 9:05 am

Posted by Brian Smith on 2004-03-25 09:00:56