The altitudes $BB_1$ and $CC_1$, are drawn in an acute triangle $ABC$. Let $X$ and $Y$ be the points, which are symmetrical to the points $B_1$ and $C_1$, with respect to the midpoints of the sides$ AB$ and $AC$ of the triangle $ABC$ respectively. Let's denote with $Z$ the point of intersection of the lines $BC$ and $XY$. Prove that the line $ZA$ is tangent to the circumscribed circle of the triangle $AXY$ .

A circle with center in $O$ is given. Two parallel tangents tangent to the circle at points $M$ and $N$. Another tangent intersects the first two tangents at points $K$ and $L$. Show that the circle having the line segment $KL$ as diameter passes through $O$.

The tangent to the circle circumscribed to the triangle $ABC$, taken through the vertex $A$, intersects the line $BC$ at the point $P$, and the tangents to the same circle, taken through $B$ and $C$, intersect the lines $AC$ and $AB$, respectively at the points $Q$ and $R$. Prove that the points $P, Q$ ¸ and $R$ are collinear.

Points $O_1$ and $O_2$ are the centers of the circumscribed and inscribed circles of an isosceles triangle $ABC$ ($AB = BC$). The circumcircles of triangles $ABC$ and $O_1O_2A$ intersect at points $A$ and $D$. Prove that line $BD$ is tangent to the circumcircle of the triangle $O_1O_2A$.

Let \( ABC \) be an acute-angled scalene triangle with circumcenter \( O \). Denote by \( M \), \( N \), and \( P \) the midpoints of sides \( BC \), \( CA \), and \( AB \), respectively. Let \( \omega \) be the circle passing through \( A \) and tangent to \( OM \) at \( O \). The circle \( \omega \) intersects \( AB \) and \( AC \) at points \( E \) and \( F \), respectively (where \( E \) and \( F \) are distinct from \( A \)). Let \( I \) be the midpoint of segment \( EF \), and let \( K \) be the intersection of lines \( EF \) and \( NP \). Prove that \( AO = 2IK \) and that triangle \( IMO \) is isosceles.

A (convex) trapezoid $ABCD$ is good, if it is inscribed in a circle, sides $AB$ and $CD$ are the bases and $CD$ is shorter than $AB$. For a good trapezoid $ABCD$ the following terms are defined: $\bullet$ The parallel to $AD$ passing through $B$ intersects the extension of side $CD$ at point $S$. $\bullet$ The two tangents passing through $S$ on the circumircle of the trapezoid touch the circle at $E$ and $F$, where $E$ lies on the same side of the straight line $CD$ as $A$. Give the simplest possible equivalent condition (expressed in side lengths and / or angles of the trapezoid) so that with a good trapezoid $ABCD$ the two angles $\angle BSE$ and $\angle FSC$ have the same measure. (Walther Janous)

Let $ABC$ be a triangle inscribed in circle $(O)$ such that points $B, C$ are fixed, while $A$ moves on major arc $BC$ of $(O)$. The tangents through $B$ and $C$ to $(O)$ intersect at $P$. The circle with diameter $OP$ intersects $AC$ and $AB$ at $D$ and $E$, respectively. Prove that $DE$ is tangent to a fixed circle whose radius is half the radius of $(O)$.

A circle touches the side $AB$ of the triangle $ABC$ at $A$, touches the side $BC$ at $P$ and intersects the side $AC$ at $Q$. The line symmetrical to $PQ$ with respect to $AC$ meets the line $AP$ at $X$. Prove that $PC=CX$. [i]Proposed by S. Berlov[/i]

For $\triangle ABC$ and its circumcircle $\omega$, draw the tangents at $B,C$ to $\omega$ meeting at $D$. Let the line $AD$ meet the circle with center $D$ and radius $DB$ at $E$ inside $\triangle ABC$. Let $F$ be the point on the extension of $EB$ and $G$ be the point on the segment $EC$ such that $\angle AFB=\angle AGE=\angle A$. Prove that the tangent at $A$ to the circumcircle of $\triangle AFG$ is parallel to $BC$. [i]Proposed by 61plus[/i]

For a nonisosceles triangle $ABC$, consider the altitude from vertex $A$ and two bisectrices from remaining vertices. Prove that the circumcircle of the triangle formed by these three lines touches the bisectrix from vertex $A$.

Given triangle $ABC$. Point $B_1$ is marked on line $AC$ so that $AB = AB_1$, while $B_1$ and $C$ are on the same side of $A$. Through points $C$, $B_1$ and the foot of the bisector of angle $A$ of triangle $ABC$, a circle $\omega$ is drawn, intersecting for second time the circle circumscribed around triangle $ABC$, at point $Q$. Prove that the tangent drawn to $\omega$ at point $Q$ is parallel to $AC$.

A circle $\omega$ and a point $T$ outside the circle is given. Let a tangent from $T$ to $\omega$ touch $\omega$ at $A$, and take points $B,C$ lying on $\omega$ such that $T,B,C$ are colinear. The bisector of $\angle ATC$ intersects $AB$ and $AC$ at $P$ and $Q$,respectively. Prove that $PA=\sqrt{PB\cdot QC}$

Let $\vartriangle ABC$ be a triangle with circumcenter $O$. Let $ P$ and $Q$ be internal points on the sides $AB$ and $AC$ respectively such that $\angle POB = \angle ABC$ and $\angle QOC = \angle ACB$. Show that the reflection of line $BC$ over line $PQ$ is tangent to the circumcircle of triangle $\vartriangle APQ$.

Let $ABC$ be an isosceles triangle with $AB=AC$. Circle $\Gamma$ lies outside $ABC$ and touches line $AC$ at point $C$. The point $D$ is chosen on circle $\Gamma$ such that the circumscribed circle of the triangle $ABD$ touches externally circle $\Gamma$. The segment $AD$ intersects circle $\Gamma$ at a point $E$ other than $D$. Prove that $BE$ is tangent to circle $\Gamma$ .

Given a circle with center $O$ and radius equal to $1$. $AB$ and $AC$ are the tangents to this circle from point $A$. Point $M$ on the circle is such that the areas of quadrilaterals $OBMC$ and $ABMC$ are equal. Find $MA$.

A circle $k$ with center $O$ and radius $r$ and a line $p$ which has no common points with $k$, are given. Let $E$ be the foot of the perpendicular from $O$ to $p$. Let $M$ be an arbitrary point on $p$, distinct from $E$. The tangents from the point $M$ to the circle $k$ are $MA$ and $MB$. If $H$ is the intersection of $AB$ and $OE$, then prove that $OH=\frac{r^2}{OE}$.

A circle $\omega$ touches the sides $A$B and $BC$ of a triangle $ABC$ and intersects its side $AC$ at $K$. It is known that the tangent to $\omega$ at $K$ is symmetrical to the line $AC$ with respect to the line $BK$. What can be the difference $AK -CK$ if $AB = 9$ and $BC = 11$?

Points $A$ and $B$ are fixed on a circle $c_1$. Circle $c_2$, whose centre lies on $c_1$, touches line $AB$ at $B$. Another line through $A$ intersects $c_2$ at points $D$ and $E$, where $D$ lies between $A$ and $E$. Line $BD$ intersects $c_1$ again at $F$. Prove that line $EB$ is tangent to $c_1$ if and only if $D$ is the midpoint of the segment $BF$.

As shown in figure, from a point $P$ exterior of circle $\odot O$, we draw tangent $PA$ and the secant $PBC$. Let $AD \perp PO$ Prove that $AC$ is tangent to the circumcircle of $\vartriangle ABD$. [img]https://cdn.artofproblemsolving.com/attachments/a/f/32da6d4626bb3592cec19a4cf0202121ba64db.png[/img]

$AD$ and $BE$ are the altitudes of the triangle $ABC$. It turned out that the point $C'$, symmetric to the vertex $C$ wrt to the midpoint of the segment $DE$, lies on the side $AB$. Prove that $AB$ is tangent to the circle circumscribed around the triangle $DEC'$.

Let $ABCD$ be a trapezoid which is not a parallelogram, such that $AD$ is parallel to $BC$. Let $O=BD\cap AC$ and $S$ be the second intersection of the circumcircles of triangles $AOB$ and $DOC$. Prove that the circumcircles of triangles $ASD$ and $BSC$ are tangent.

A circle, its chord $AB$ and the midpoint $W$ of the minor arc $AB$ are given. Take an arbitrary point $C$ on the major arc $AB$. The tangent to the circle at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$ respectively. Lines $WX$ and WY meet AB at points $N$ and $M$ respectively. Prove that the length of segment $NM$ does not depend on point $C$. (A. Zertsalov, D. Skrobot)

Show that there are real numbers $A,B$ such that the identity \[\sum_{k=1}^n\tan(k)\tan(k-1)=A\tan(n)+Bn\] holds for every positive integer $n.$

The quadrilateral $ABCD$ is known to be inscribed in a circle, and that there is a circle with center on side $AD$ tangent to the other three sides. Prove that $AD = AB + CD$.

Given $\triangle ABC$ with $AB < AC$, let $\omega$ be the circle centered at the midpoint $M$ of $BC$ with diameter $AC - AB$. The internal bisector of $\angle BAC$ intersects $\omega$ at distinct points $X$ and $Y$. Let $T$ be the point on the plane such that $TX$ and $TY$ are tangent to $\omega$. Prove that $AT$ is perpendicular to $BC$.