Found problems: 5
Let $ABC$ be a non-right-angled triangle, with $AC\ne BC$. Let $F$ be the midpoint of side $BC$. Let $D$ be a point on line $AB$ satisfying$CA=CD$,and let $E$ be a point on line $BC$ satisfying $EB = ED$. The line passing through $A$ and parallel to $ED$ meets line $FD$ at point $I$. Line $AF$ meets line $ED$ at point $J$. Prove that points $C$, $I$ and $J$ are collinear.
Given a parallelogram $ABCD$. The line perpendicular to $AC$ passing through $C$ and the line perpendicular to $BD$ passing through $A$ intersect at point $P$. The circle centered at point $P$ and radius $PC$ intersects the line $BC$ at point $X$, ($X \ne C$) and the line $DC$ at point $Y$ , ($Y \ne C$). Prove that the line $AX$ passes through the point $Y$ .
Circle $\omega$ is inscribed in quadrilateral $ABCD$ such that $AB$ and $CD$ are not parallel and
intersect at point $O.$ Circle $\omega_1$ touches the side $BC$ at $K$ and touches line $AB$ and $CD$ at
points which are located outside quadrilateral $ABCD;$ circle $\omega_2$ touches side $AD$ at $L$ and
touches line $AB$ and $CD$ at points which are located outside quadrilateral $ABCD.$ If $O,K,$
and $L$ are collinear$,$ then show that the midpoint of side $BC,AD,$ and the center of circle
$\omega$ are also collinear.
Let $ABC$ be a triangle, such that the midpoint of $AB$, the incenter and the touchpoint of the excircle opposite $A$ with $\overline{AC}$ are collinear. Find $AB$ and $BC$ if $AC=3$ and $\angle ABC=60^{\circ}$.
Circle $\omega$ is inscribed in quadrilateral $ABCD$ such that $AB$ and $CD$ are not parallel and
intersect at point $O.$ Circle $\omega_1$ touches the side $BC$ at $K$ and touches line $AB$ and $CD$ at
points which are located outside quadrilateral $ABCD;$ circle $\omega_2$ touches side $AD$ at $L$ and
touches line $AB$ and $CD$ at points which are located outside quadrilateral $ABCD.$ If $O,K,$
and $L$ are collinear$,$ then show that the midpoint of side $BC,AD,$ and the center of circle
$\omega$ are also collinear.