BG-MO 2014

Let $k$ be a given circle and $A$ is a fixed point outside $k$ . $BC$ is a diameter of $k$ . Find the locus of the orthocentre of $\triangle ABC$ when $BC$ varies.

Proposed by T. Vitanov, E. Kolev

Let $ABCD$ be a quadrilateral inscribed in a circle $k$ . $AC$ and $BD$ meet at $E$ . The rays $\overrightarrow{CB}, \overrightarrow{DA}$ meet at $F$ . Prove that the line through the incenters of $\triangle ABE\,, \triangle ABF$ and the line through the incenters of $\triangle CDE\,, \triangle CDF$ meet at a point lying on the circle $k$ .

Proposed by N. Beluhov