2005 Cyprus Seniors TST/Day 1/Problem 3

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Problem

Given a circle with ceneter $O$ and an inscribed trapezium $AB\Gamma\Delta$ $(AB//\Gamma\Delta)$ with $AB<\Gamma\Delta$. If $P$ is a point of arc $\Gamma\Delta$ on which $A$ and $B$ do not belong to, and $P_{1},P_{2},P_{3}$ and $P_{4}$ are the projections of $R$ on the lines $A\Delta$, $B\Gamma$, $B\Delta$, and $A\Gamma$ respectively, show that:

(i) The circumscribed circles of the triangles $PP_{1}\Delta$ and $PP_{2}\Gamma$ intersect on the side $\Gamma\Delta$.

(ii) The points $P_{1},P_{2},P_{3}$ and $P_{4}$ are concyclic.

(iii) If $AB=\alpha$, $\Gamma\Delta=\beta$ and the distance between the parallel chords is $h$ find all the points of the axis of symmetry of the trapezium $AB\Gamma\Delta$ that can `see'* at right angle the non parallel sides and calculate their distance form $AB$ and $\Gamma\Delta$ in terms of $\alpha$, $\beta$ and $h$. Examinate if such pints always exist. \newline (Draw a separate diagram for part (iii)).

*An example what I mean: In any cyclic quadrilateral $ABCD$ the point $A$ and $B$ `see' the side $CD$ at the same angle.


Solution


See also