Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.7 AIME Prove it Greece

Let $\triangle AB\Gamma$ be an acute angled triangle with circumcircle $c(O, R)$ . From the midpoint $\Delta$ of the side $B\Gamma$ we draw a line perpendicular to $AB$ which meets $AB$ at $E$ . If the line $AO$ intersects the line $\varepsilon$ at $Z$ , prove that the points $A$ , $Z$ , $\Delta$ , $\Gamma$ are cyclic.

Solution

The external angle $\mathrm{E}\hat{Z}A$ of the quadrilateral $AZ\Delta\Gamma$ belongs to the orthogonal triangle $AEZ$ , with the acute angle $\mathrm{E}\hat{A}Z = \omega$ equal to the angle $A\hat{B}O$ , since $OA = OB$ . Hence $\mathrm{E}\hat{Z}A = 90^\circ - \omega$

Let the extension of the radius $BO$ intersect the circle $c(O, R)$ at $H$ . Then $A\hat{\Gamma}H = A\hat{B}H = \omega$ and
$90^\circ = B\hat{\Gamma}H = B\hat{\Gamma}A + A\hat{\Gamma}H \Rightarrow B\hat{\Gamma}A = 90^\circ - A\hat{\Gamma}H = 90^\circ - \omega.$
Hence $\mathrm{E}\hat{Z}A = B\hat{\Gamma}A$ , and the quadrilateral $AZ\Delta\Gamma$ is cyclic.

Figure 4

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