In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

Given an acute triangle $ABC$ ($AC\ne AB$) and let $(C)$ be its circumcircle. The excircle $(C_1)$ corresponding to the vertex $A$, of center $I_a$, tangents to the side $BC$ at the point $D$ and to the extensions of the sides $AB,AC$ at the points $E,Z$ respectively. Let $I$ and $L$ are the intersection points of the circles $(C)$ and $(C_1)$, $H$ the orthocenter of the triangle $EDZ$ and $N$ the midpoint of segment $EZ$. The parallel line through the point $l_a$ to the line $HL$ meets the line $HI$ at the point $G$. Prove that the perpendicular line $(e)$ through the point $N$ to the line $BC$ and the parallel line $(\delta)$ through the point $G$ to the line $IL$ meet each other on the line $HI_a$.

In a triangle $ABC$, the nine-point circle $(N)$ is tangent to the incircle $(I)$ and three excircles $(I_a), (I_b), (I_c)$ at the Feuerbach points $F, F_a, F_b, F_c$. Tangents of $(N)$ at $F, F_a, F_b, F_c$ bound a quadrangle $PQRS$. Show that the Euler line of $ABC$ is a Newton line of $PQRS$. Luis González

Let $ABC$ be a triangle with $\angle ABC$ obtuse. The [i]$A$-excircle[/i] is a circle in the exterior of $\triangle ABC$ that is tangent to side $BC$ of the triangle and tangent to the extensions of the other two sides. Let $E$, $F$ be the feet of the altitudes from $B$ and $C$ to lines $AC$ and $AB$, respectively. Can line $EF$ be tangent to the $A$-excircle? [i]Proposed by Ankan Bhattacharya, Zack Chroman, and Anant Mudgal[/i]

Let $I$ be the incenter of an acute triangle $ABC$. Assume that $K_1$ is the point such that $AK_1 \perp BC$ and the circle with center $K_1$ of radius $K_1A$ is internally tangent to the incircle of triangle $ABC$ at $A_1$. The points $B_1, C_1$ are defined similarly. a) Prove that $AA_1, BB_1, CC_1$ are concurrent at a point $P$. b) Let $\omega_1,\omega_2,\omega_3$ be the excircles of triangle $ABC$ with respect to $A, B, C$, respectively. The circles $\gamma_1,\gamma_2\gamma_3$ are the reflections of $\omega_1,\omega_2,\omega_3$ with respect to the midpoints of $BC, CA, AB$, respectively. Prove that P is the radical center of $\gamma_1,\gamma_2,\gamma_3$.

It turned out for some triangle with sides $a, b$ and $c$, that a circle of radius $r = \frac{a+b+c}{2}$ touches side $c$ and extensions of sides $a$ and $b$. Prove that a circle of radius $ \frac{a+c-b}{2}$ is tangent to $a$ and the extensions of $b$ and $c$.

Let be two fixed points $ B,C. $ Find the locus of the spatial points $ A $ such that $ ABC $ is a nondegenerate triangle and the expression $$ R^2 (A)\cdot\sin \left( 2\angle ABC\right)\cdot\sin \left( 2\angle BCA\right) $$ has the greatest value possible, where $ R(A) $ denotes the radius of the excirlce of $ ABC. $

Let $ABC$ be a non-isosceles triangle with circumcircle $k$, incenter $I$ and $C$-excenter $I_C$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of arc $\widehat{ACB}$ on $k$. Prove that $\angle IMI_C + \angle INI_C = 180^{\circ}$.

Let $M$ be the midpoint of side BC of an acute-angled triangle $ABC$. Let $D$ and $E$ be the center of the excircle of triangle $AMB$ tangent to side $AB$ and the center of the excircle of triangle $AMC$ tangent to side $AC$, respectively. The circumscribed circle of triangle $ABD$ intersects line$ BC$ for the second time at point $F$, and the circumcircle of triangle $ACE$ is at point $G$. Prove that $| BF | = | CG|$.

Let $ABC$ be an acute scalene triangle. Its $C$-excircle tangent to the segment $AB$ meets $AB$ at point $M$ and the extension of $BC$ beyond $B$ at point $N$. Analogously, its $B$-excircle tangent to the segment $AC$ meets $AC$ at point $P$ and the extension of $BC$ beyond $C$ at point $Q$. Denote by $A_1$ the intersection point of the lines $MN$ and $PQ$, and let $A_2$ be defined as the point, symmetric to $A$ with respect to $A_1$. Define the points $B_2$ and $C_2$, analogously. Prove that $\triangle ABC$ is similar to $\triangle A_2B_2C_2$.

Let $ABC$ be a triangle with circumcircle $\omega$ and let $\Omega_A$ be the $A$-excircle. Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_A$. Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_A$ at $X$ and $Y$ respectively. The tangent line at $P$ to the circumcircle of the triangle $APX$ intersects the tangent line at $Q$ to the circumcircle of the triangle $AQY$ at a point $R$. Prove that $\overline{AR} \perp \overline{BC}$.

Let $ABC$ be the triangle in which $AB> AC$. Circle $\omega_a$ touches the segment of the $BC$ at point $D$, the extension of the segment $AB$ towards point $B$ at the point $F$, and the extension of the segment $AC$ towards point $C$ at the point $E$. The ray $AD$ intersects circle $\omega_a$ for second time at point $M$. Denote the circle circumscribed around the triangle $CDM$ by $\omega$. Circle $\omega$ intersects the segment $DF$ at N. Prove that $FN > ND$.

Let $\omega$ be the $A$-excircle of triangle $ABC$ and $M$ the midpoint of side $BC$. $G$ is the pole of $AM$ w.r.t $\omega$ and $H$ is the midpoint of segment $AG$. Prove that $MH$ is tangent to $\omega$.

A triangle has sides $a, b,c$, radius of the incircle $r$ and radii of the excircles $r_a, r_b, r_c$: Prove that: a) The triangle is right-angled if and only if: $r + r_a + r_b + r_c = a + b + c$. b) The triangle is right-angled if and only if: $r^2 + r^2_a + r^2_b + r^2_c = a^2 + b^2 + c^2$.

Let $ABC$ be a triangle, with $A'$, $B'$, and $C'$ the points of tangency of the incircle with $BC$, $CA$, and $AB$ respectively. Let $X$ be the intersection of the excircle with respect to $A$ with $AB$, and $M$ the midpoint of $BC$. Let $D$ be the intersection of $XM$ with $B'C'$. Show that $\angle C'A'D' = 90^o$.

In triangle $ABC$ with circumcircle $\Omega$ and incenter $I$, point $M$ bisects arc $BAC$ and line $\overline{AI}$ meets $\Omega$ at $N\ne A$. The excircle opposite to $A$ touches $\overline{BC}$ at point $E$. Point $Q\ne I$ on the circumcircle of $\triangle MIN$ is such that $\overline{QI}\parallel\overline{BC}$. Prove that the lines $\overline{AE}$ and $\overline{QN}$ meet on $\Omega$.

Let $ABC$ be a triangle. The ex-circles touch sides $BC, CA$ and $AB$ at points $U, V$ and $W$, respectively. Be $r_u$ a straight line that passes through $U$ and is perpendicular to $BC$, $r_v$ the straight line that passes through $V$ and is perpendicular to $AC$ and $r_w$ the straight line that passes through W and is perpendicular to $AB$. Prove that the lines $r_u$, $r_v$ and $r_w$ pass through the same point.

Given a right triangle $ABC$ ($\angle C=90^o$). The $C$-excircle touches the hypotenuse $AB$ at point $C_1, A_1$ is the touchpoint of $B$-excircle with line $BC, B_1$ is the touchpoint of $A$-excircle with line $AC$. Find the angle $\angle A_1C_1B_1$.

In the triangle $ABC$, the segment $CL$ is the angle bisector. The $C$-exscribed circle with center at the point $ I_c$ touches the side of the $AB$ at the point $D$ and the extension of sides $CA$ and $CB$ at points $P$ and $Q$, respectively. It turned out that the length of the segment $CD$ is equal to the radius of this exscribed circle. Prove that the line $PQ$ bisects the segment $I_CL$.

Prove that a circle constructed with the side $AB$ of a triangle $ABC$ as a diameter touches the inscribed circle of the triangle $ABC$ if and only if the side $AB$ is equal to the radius of the exircle on that side.

The circle $\Omega$ with center $I_A$, is the $A$-excircle of triangle $ABC$. Which is tangent to $AB,AC$ at $F,E$ respectivly. Point $D$ is the reflection of $A$ through $I_AB$. Lines $DI_A$ and $EF$ meet at $K$. Prove that ,circumcenter of $DKE$ , midpoint of $BC$ and $I_A$ are collinear.

Let $\Omega$ be the $A$-excircle of triangle $ABC$, and suppose that $\Omega$ is tangent to lines $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Let $M$ be the midpoint of segment $EF$. Two more points $P$ and $Q$ are on $\Omega$ such that $EP$ and $FQ$ are both parallel to $DM$. Let $BP$ meet $CQ$ at point $X$. Prove that the line $AM$ is the angle bisector of $\angle XAD$. [i]Proposed by Shuang-Yen Lee[/i]

Let $k$ be the circumcircle of $\triangle ABC$ and let $k_a$ be A-excircle .Let the two common tangents of $k,k_a$ cut $BC$ in $P,Q$.Prove that $\measuredangle PAB=\measuredangle CAQ$.

Let $ABC$ be a triangle with $AB \neq AC$, let $I$ be its incenter, $\gamma$ its inscribed circle and $D$ the midpoint of $BC$. The tangent to $\gamma$ from $D$ different to $BC$ touches $\gamma$ in $E$. Prove that $AE$ and $DI$ are parallel.

Consider a triangle $ABC$ in which $AB<BC<CA$. The excircles touch the sides $BC, CA,$ and $AB$ at points $A_1, B_1$ and $C_1$ respectively. A circle is drawn through the points $A_1, B_1$ and $C_1$ which intersects the sides $BC, CA$ and $AB$ for the second time at the points $A_2, B_2$ and $C_2$ respectively. On which side of the triangle can lie the largest of the segments $A_1A_2, B_1B_2$ and $C_1C_2$? [i]Proposed by I. Weinstein[/i]