Baltic Way 2011 · Shortlist problem

Geometry

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

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Topics

Angles and distances · Constructions, loci, concurrency and collinearity

Solutions

Solution

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

Diagram for the mathnet 0190 1 of bw-cand-2011-mn-0190.

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