Found problems: 353
Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and \[
\angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \] Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.
In a non-isosceles triangle $ABC$ the centre of the incircle is denoted by $I$. The other intersection point of the angle bisector of $\angle BAC$ and the circumcircle of $\vartriangle ABC$ is $D$. The line through $I$ perpendicular to $AD$ intersects $BC$ in $F$. The midpoint of the circle arc $BC$ on which $A$ lies, is denoted by $M$. The other intersection point of the line $MI$ and the circle through $B, I$ and $C$, is denoted by $N$. Prove that $FN$ is tangent to the circle through $B, I$ and $C$.
Let $ABCD$ be a cyclic quadrilateral and let $k_1$ and $k_2$ be circles inscribed in triangles $ABC$ and $ABD$. Prove that external common tangent of those circles (different from $AB$) is parallel with $CD$
In triangle $ABC$, point $M$ is the midpoint of $BC$, $P$ the point of intersection of the tangents at points $B$ and $C$ of the circumscribed circle of $ABC$, $N$ is the midpoint of the segment $MP$. The segment $AN$ meets the circumcircle $ABC$ at the point $Q$. Prove that $\angle PMQ = \angle MAQ$.
Circles $\omega_1$ and $\omega_2$ inscribed into equal angles $X_1OY$ and $Y OX_2$ touch lines $OX_1$ and $OX_2$ at points $A_1$ and $A_2$ respectively. Also they touch $OY$ at points $B_1$ and $B_2$. Let $C_1$ be the second common point of $A_1B_2$ and $\omega_1, C_2$ be the second common point of $A_2B_1$ and $\omega_2$. Prove that $C_1C_2$ is the common tangent of two circles.
A circle inscribed in an angle with vertex $O$ touches its sides at points $A$ and $B$, $K$ is an arbitrary point on the smaller of the two arcs $AB$ of this circle. On the line $OB$ a point $L$ is taken such that the lines $OA$ and $KL$ are parallel. Let $M$ be the intersection point of the circle $\omega$ circumscribed around triangle $KLB$, with line $AK$, with $M$ different from $K$. Prove that line $OM$ touches circle $\omega$.
Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$.
[i]Proposed by David Monk, United Kingdom[/i]
Let $ABC$ be an isosceles triangle with $AB = AC$. The points $D, E$ and $F$ are taken on the sides $BC, CA$ and $AB$, respectively, so that $\angle F DE = \angle ABC$ and $FE$ is not parallel to $BC$. Prove that the line $BC$ is tangent to the circumcircle of $\vartriangle DEF$ if and only if $D$ is the midpoint of the side $BC$.
Let $I$ be the incenter of a non-isosceles triangle $ABC$. The line $AI$ intersects the circumcircle of the triangle $ABC$ at $A$ and $D$. Let $M$ be the middle point of the arc $BAC$. The line through the point $I$ perpendicular to $AD$ intersects $BC$ at $F$. The line $MI$ intersects the circle $BIC$ at $N$.
Prove that the line $FN$ is tangent to the circle $BIC$.
Let $C$ be a circle, and let $A, B, P$ be three points on $C$. Let $L_A$ and $L_B$ be the tangent lines to $C$ passing through $A$ and $B$, respectively. Let $a$ and $b$ be the distances from $P$ to $L_A$ and $L_B$, respectively, and let $c$ be the distance from $P$ to the chord of $C$ determined by $A$ and $B$. Prove that $c^2 = a \cdot b$.
Two equal circles $S_1$ and $S_2$ meet at two different points. The line $\ell$ intersects $S_1$ at points $A,C$ and $S_2$ at points $B,D$ respectively (the order on $\ell$: $A,B,C,D$) . Define circles $\Gamma_1$ and $\Gamma_2$ as follows: both $\Gamma_1$ and $\Gamma_2$ touch $S_1$ internally and $S_2$ externally, both $\Gamma_1$ and $\Gamma_2$ line $\ell$, $\Gamma_1$ and $\Gamma_2$ lie in the different halfplanes relatively to line $\ell$. Suppose that $\Gamma_1$ and $\Gamma_2$ touch each other. Prove that $AB=CD$.
I. Voronovich
Inside the inscribed quadrilateral $ ABCD $, a point $ P $ is marked such that $ \angle PBC = \angle PDA $, $ \angle PCB = \angle PAD $. Prove that there exists a circle that touches the straight lines $ AB $ and $ CD $, as well as the circles circumscribed by the triangles $ ABP $ and $ CDP $.
Given a cyclic quadrilateral $ABCD$, define $E$ as $AD \cap BC$ and $F$ as $AB \cap CD$.
Let $\Omega_A$ be the circle passing through $A, D$ and tangent to $AB$, and let its center be $O_A$.
Let $\Gamma_B$ be the circle passing through $B, C$ and tangent to $AB$, and let its center be $O_B$.
Let $\Gamma_C$ be the circle passing through $B, C$ and tangent to $CD$, and let its center be $O_C$.
Let $\Omega_D$ be the circle passing through $A, D$ and tangent to $CD$, and let its center be $O_D$.
Prove that $O_AO_BO_CO_D$ is cyclic, and prove that it's center lies on $EF$.
Let $AM$ and $AN$ be the lines tangent to a circle $\Gamma$ drawn from a point $A$ $(M$ and $N$ belong to the circle). A line through $A$ cuts $\Gamma$ at $B$ and $C$ with $B$ between $A$ and $C$, and $\frac{AB}{BC} =\frac23$. If $P$ is the intersection point of $AB$ and $MN$, calculate $\frac{AP}{CP}$.
In Fig. , $PA$ and $QB$ are tangents to the circle at $A$ and $B$ respectively. The line $AB$ is extended to meet $PQ$ at $S$. Suppose that $PA = QB$. Prove that $QS = SP$.
[img]https://cdn.artofproblemsolving.com/attachments/6/f/f21c0c70b37768f3e80e9ee909ef34c57635d5.png[/img]
Two circles $\alpha$ and $\beta$ intersect at points $P$ and $Q$. The lines connecting a point $R$ on $\beta$ with $P$ and $Q$ intersect $\alpha$ again at $S$ and $T$ respectively. Prove that $ST$ is parallel to the line tangent to $\beta$ at $R$.
$C$ is a point on the semicircle diameter $AB$, between $A$ and $B$. $D$ is the foot of the perpendicular from $C$ to $AB$. The circle $K_1$ is the incircle of $ABC$, the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$.
Let $A$ be a point outside the circle $\omega$. Two points $B, C$ lie on $\omega$ such that $AB, AC$ are tangent to $\omega$. Let $D$ be any point on $\omega$ ($D$ is neither $B$ nor $C$) and $M$ the foot of perpendicular from $B$ to $CD$. The line through $D$ and the midpoint of $BM$ meets $\omega$ again at $P$. Prove that $AP \perp CP$
Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$.
[i]Proposed by David Monk, United Kingdom[/i]
On the plane is drawn a triangle $ABC$ and a circle $\omega$ passing through the vertex $C$, the midpoints of the sides $AC$ and $BC$ and the point of intersection of the medians of the triangle $ABC$. The point $K$ lies on the circle $\omega$ such that $\angle AKB=90^o$. Using only with a ruler, draw a tangent to the circle $\omega$ at point $K$.
Let $CX, CY$ be the tangents from vertex $C$ of triangle $ABC$ to the circle passing through the midpoints of its sides. Prove that lines $XY , AB$ and the tangent to the circumcircle of $ABC$ at point $C$ concur.
Given an $ABC$ acute triangle with $O$ the center of the circumscribed circle. Suppose that $\omega$ is a circle that is tangent to the line $AO$ at point $A$ and also tangent to the line $BC$. Prove that $\omega$ is also tangent to the circumcircle of the triangle $BOC$.
A circle and a point $P$ inside it are given. Two arbitrary perpendicular rays starting at point $P$ intersect the circle at points $A$ and $B$. Point $X$ is the projection of point $P$ onto line $AB, Y$ is the intersection point of tangents to the circle drawn through points $A$ and $B$. Prove that all lines $XY$ pass through the same point.
(A. Zaslavsky)
A teacher gives each student in the class the following task in their exercise book .
"Take two concentric circles of radius $1$ and $10$ . To the smaller circle produce three tangents whose intersections $A, B$ and $C$ lie in the larger circle . Measure the area $S$ of triangle $ABC$, and areas $S_1 , S_2$ and $S_3$ , the three sector-like regions with vertices at $A, B$ and $C$ (as in the diagram). Find the value of $S_1 +S_2 +S_3 -S$."
Prove that each student would obtain the same result .
[img]https://1.bp.blogspot.com/-K3kHWWWgxgU/XWHRQ8WqqPI/AAAAAAAAKjE/0iO4-Yz6p9AcM2mklprX_M18xTyg9O81gCK4BGAYYCw/s200/TOT%2B1985%2BAutumn%2BJ3.png[/img]
( A . K . Tolpygo, Kiev)
Let $ABCD$ be a trapezoid with $AD \parallel BC$ and $\angle BCD < \angle ABC < 90^\circ$. Let $E$ be the intersection point of the diagonals $AC$ and $BD$. The circumcircle $\omega$ of $\triangle BEC$ intersects the segment $CD$ at $X$. The lines $AX$ and $BC$ intersect at $Y$, while the lines $BX$ and $AD$ intersect at $Z$. Prove that the line $EZ$ is tangent to $\omega$ iff the line $BE$ is tangent to the circumcircle of $\triangle BXY$.