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, 2023

Geometry Difficulty 5.0 AIME, harder Prove it United Kingdom

$ABCD$ is a quadrilateral, with vertices labelled in anti-clockwise order, such that: $AB$ is parallel to $DC$ , $AB = AC$ , and angle $ADC$ is equal to angle $ACB$ .

(a) Draw a diagram to show this information. Your diagram need not be to scale, but you should mark clearly equal lengths and angles.

(b) (i) Prove that $AD = BC$ and that $AD$ is parallel to $BC$ .
(ii) What type of quadrilateral is $ABCD$ ?

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