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Geometry Difficulty 7.7 National Olympiad, round 2 Prove it Baltic Way

Suppose that the quadrilateral $ABCD$ satisfies $\angle ABD = 30^\circ$ , $\angle CDB = 20^\circ$ and $\angle BCA = \angle ACD = 40^\circ$ . Determine $\angle DAC$ .

Solution

Let $E$ be the intersection point of the diagonals $AC$ and $BD$ . Let $EF$ be the bisector of $\angle DEC$ , with $F$ on $DC$ .

Then the triangles $BEC$ and $CFE$ are congruent ( $EFCB$ is a kite). So $EF = EB = EA$ , and $ADFE$ is a kite. It follows that $\angle DAC = 100^\circ$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.