Found problems: 15
Let $\omega$ be the circumscribed circle of triangle $ABC$, and let $\omega'$ 'be the circle tangent to the side $BC$ and the extensions of the sides $AB$ and $AC$. The common tangents to the circles $\omega$ and $\omega'$ intersect the line $BC$ at points $D$ and $E$. Prove that $\angle BAD = \angle CAE$.
Let $ABC$ be a triangle with incircle $\gamma$. The circle through $A$ and $B$ tangent to $\gamma$ touches it at $C_2$ and the common tangent at $C_2$ intersects $AB$ at $C_1$. Define the points $A_1$, $B_1$, $A_2$, $B_2$ analogously. Prove that:
a) the points $A_1$, $B_1$, $C_1$ are collinear;
b) the lines $AA_2$, $BB_2$, $CC_2$ are concurrent.
Let $ABC$ be a triangle with a circumcircle $(K)$. A circle touching the sides $AB,AC$ is internally tangent to $(K)$ at $K_a$; two other points $K_b,K_c$ are defined in the same manner. Prove that the area of triangle $K_aK_bK_c$ does not exceed that of triangle $ABC$.
Nguyen Minh Ha, Hanoi University of Education, Xuan Thuy, Cau Giay, Hanoi.
Let $h$ be a semicircle with diameter $AB$. Let $P$ be an arbitrary point inside the diameter $AB$. The perpendicular through $P$ on $AB$ intersects $h$ at point $C$. The line $PC$ divides the semicircular area into two parts. A circle will be inscribed in each of them that touches $AB, PC$ and $h$. The points of contact of the two circles with $AB$ are denoted by $D$ and $E$, where $D$ lies between $A$ and $P$. Prove that the size of the angle $DCE$ does not depend on the choice of $P$.
(Walther Janous)
Let $(O)$ be the circumcircle of triangle $ABC$, and $P$ a point on the arc $BC$ not containing $A$. $(Q)$ is the $A$-mixtilinear circle of triangle $ABC$, and $(K), (L)$ are the $P$-mixtilinear circles of triangle $PAB, PAC$ respectively. Prove that there is a line tangent to all the three circles $(Q), (K)$ and $(L)$.
Nguyen Van Linh, a student at Hanoi Foreign Trade University Cabinet
Triangle $\Delta ABC$ is inscribed in circle $\Omega$. The tangency point of $\Omega$ and the $A$-mixtilinear circle of $\Delta ABC$ is $T$. Points $E$, $F$ were chosen on $AC$, $AB$ respectively so that $EF\parallel BC$ and $(TEF)$ is tangent to $\Omega$. Let $\omega$ denote the $A$-excircle of $\Delta AEF$, which is tangent to sides $EF$, $AE$, $AF$ at $K$, $Y$, $Z$ respectively. Line $AT$ intersects $\omega$ at two points $P$, $Q$ with $P$ between $A$ and $Q$. Let $QK$ and $YZ$ intersect at $V$, and let the tangent to $\omega$ at $P$ and the tangent to $\Omega$ at $T$ intersect at $U$. Prove that $UV\parallel BC$.
Let $ABC$ be a triangle. Consider the circle $\omega_B$ internally tangent to the sides $BC$ and $BA$, and to the circumcircle of the triangle $ABC$, let $P$ be the point of contact of the two circles. Similarly, consider the circle $\omega_C$ internally tangent to the sides $CB$ and $CA$, and to the circumcircle of the triangle $ABC$, let $Q$ be the point of contact of the two circles. Show that the incentre of the triangle $ABC$ lies on the segment $PQ$ if and only if $AB + AC = 3BC$.
proposed by Luis Eduardo Garcia Hernandez, Mexico
Let $ABC$ be a triangle with circumcircle $\Omega$. Let $S_b$ and $S_c$ respectively denote the midpoints of the arcs $AC$ and $AB$ that do not contain the third vertex. Let $N_a$ denote the midpoint of arc $BAC$ (the arc $BC$ including $A$). Let $I$ be the incenter of $ABC$. Let $\omega_b$ be the circle that is tangent to $AB$ and internally tangent to $\Omega$ at $S_b$, and let $\omega_c$ be the circle that is tangent to $AC$ and internally tangent to $\Omega$ at $S_c$. Show that the line $IN_a$, and the lines through the intersections of $\omega_b$ and $\omega_c$, meet on $\Omega$.
Let $\triangle{ABC}$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Incircle $(I)$ of the $\triangle{ABC}$ is tangent to the sides $BC,CA,AB$ at $M,N,P$ respectively. Denote $\Omega_A$ to be the circle passing through point $A$, external tangent to $(I)$ at $A'$ and cut again $AB,AC$ at $A_b,A_c$ respectively. The circles $\Omega_B,\Omega_C$ and points $B',B_a,B_c,C',C_a,C_b$ are defined similarly.
$a)$ Prove $B_cC_b+C_aA_c+A_bB_a \ge NP+PM+MN$.
$b)$ Suppose $A',B',C'$ lie on $AM,BN,CP$ respectively. Denote $K$ as the circumcenter of the triangle formed by lines $A_bA_c,B_cB_a,C_aC_b.$ Prove $OH//IK$.
In a circle $S$, a chord $AB$ is drawn and let $M$ be the midpoint of the arc $AB$. Let $P$ be a point in segment $AB$ other than its midpoint. The extension of the segment $MP$ cuts $S$ in $Q$. Let $S_1$ be the circle that is tangent to the AP segments and $MP$, and also is tangent to $S$, and let $S_2$ be the circle that is tangent to the segments $BP$ and $MP$, and also tangent to $S$. The common outer tangent lines to the circles $S_1$ and $S_2$ are cut at $C$. Prove that $\angle MQC = 90^o$.
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 $ABCD$ be a quadrilateral inscribed in a circle $k$. $AC$ and $BD$ meet at $E$. The rays $\overrightarrow{CB}, \overrightarrow{DA}$ meet at $F$.
Prove that the line through the incenters of $\triangle ABE\,,\, \triangle ABF$ and the line through the incenters of $\triangle CDE\,,\, \triangle CDF$ meet at a point lying on the circle $k$.
[i]Proposed by N. Beluhov[/i]
Given a triangle $ABC$. Let $\Omega$ be the circumscribed circle of this triangle, and $\omega$ be the inscribed circle of this triangle. Let $\delta$ be a circle that touches the sides $AB$ and $AC$, and also touches the circle $\Omega$ internally at point $D$. The line $AD$ intersects the circle $\Omega$ at two points $P$ and $Q$ ($P$ lies between $A$ and $Q$). Let $O$ and $I$ be the centers of the circles $\Omega$ and $\omega$. Prove that $OD \parallel IQ$.
Let $\omega$ be the circumscribed circle of the triangle $ABC$, in which $AC< AB$, $K$ is the center of the arc $BAC$, $KW$ is the diameter of the circle $\omega$. The circle $\gamma$ is inscribed in the curvilinear triangle formed by the segments $BC$, $AB$ and the arc $AC$ of the circle $\omega$. It turned out that circle $\gamma$ also touches $KW$ at point $F$. Let $I$ be the center of the triangle $ABC$, $M$ is the midpoint of the smaller arc $AK$, and $T$ is the second intersection point of $MI$ with the circle $\omega$. Prove that lines $FI$, $TW$ and $BC$ intersect at one point.
(Mykhailo Sydorenko)
Let $ABC$ be an acute-angled triangle \( ABC \) with \( AC > BC \) and incenter \( I \). Let \( \omega \) be the mixtilinear circle at vertex \( C \), i.e. the circle internally tangent to the circumcircle of \( \triangle ABC \) and also tangent to lines \( AC \) and \( BC \). A circle \( \Gamma \) passes through points \( A \) and \( B \) and is tangent to \( \omega \) at point \( T \), with \( C \notin \Gamma \) and \( I \) being inside \( \triangle ATB \). Prove that:
$$\angle CTB + \angle ATI = 180^\circ + \angle BAI - \angle ABI.$$