Školjka

%V0 A line in the plane of a triangle $ABC$ intersects the sides $AB$ and $AC$ respectively at points $X$ and $Y$ such that $BX = CY$ . Find the locus of the center of the circumcircle of triangle $XAY .$

%V0 Let $ABCD$ be a convex quadrilateral whose vertices do not lie on a circle. Let $A'B'C'D'$ be a quadrangle such that $A',B', C',D'$ are the centers of the circumcircles of triangles $BCD,ACD,ABD$, and $ABC$. We write $T (ABCD) = A'B'C'D'$. Let us define $A''B''C''D'' = T (A'B'C'D') = T (T (ABCD)).$ (a) Prove that $ABCD$ and $A''B''C''D''$ are similar. (b) The ratio of similitude depends on the size of the angles of $ABCD$. Determine this ratio.

%V0 The triangle $ABC$ is inscribed in a circle. The interior bisectors of the angles $A,B$ and $C$ meet the circle again at $A', B'$ and $C'$ respectively. Prove that the area of triangle $A'B'C'$ is greater than or equal to the area of triangle $ABC.$

%V0 Let $ABC$ be an acute-angled triangle. The lines $L_{A}$, $L_{B}$ and $L_{C}$ are constructed through the vertices $A$, $B$ and $C$ respectively according the following prescription: Let $H$ be the foot of the altitude drawn from the vertex $A$ to the side $BC$; let $S_{A}$ be the circle with diameter $AH$; let $S_{A}$ meet the sides $AB$ and $AC$ at $M$ and $N$ respectively, where $M$ and $N$ are distinct from $A$; then let $L_{A}$ be the line through $A$ perpendicular to $MN$. The lines $L_{B}$ and $L_{C}$ are constructed similarly. Prove that the lines $L_{A}$, $L_{B}$ and $L_{C}$ are concurrent.

%V0 Let $Q$ be the centre of the inscribed circle of a triangle $ABC.$ Prove that for any point $P,$ $$a(PA)^2 + b(PB)^2 + c(PC)^2 = a(QA)^2 + b(QB)^2 + c(QC)^2 + (a + b + c)(QP)^2,$$ where $a = BC, b = CA$ and $c = AB.$

%V0 A point $M$ is chosen on the side $AC$ of the triangle $ABC$ in such a way that the radii of the circles inscribed in the triangles $ABM$ and $BMC$ are equal. Prove that $$BM^{2} = X \cot \left( \frac {B}{2}\right)$$ where X is the area of triangle $ABC.$