2005 Cyprus Seniors TST/Day 1/Problem 3
Problem
Given a circle with ceneter and an inscribed trapezium with . If is a point of arc on which and do not belong to, and and are the projections of on the lines , , , and respectively, show that:
(i) The circumscribed circles of the triangles and intersect on the side .
(ii) The points and are concyclic.
(iii) If , and the distance between the parallel chords is find all the points of the axis of symmetry of the trapezium that can `see'* at right angle the non parallel sides and calculate their distance form and in terms of , and . Examinate if such pints always exist. \newline (Draw a separate diagram for part (iii)).
*An example what I mean: In any cyclic quadrilateral the point and `see' the side at the same angle.