In a non-isosceles triangle $ABC$,$O$ and $I$ are circumcenter and incenter,respectively.$B^\prime$ is reflection of $B$ with respect to $OI$ and lies inside the angle $ABI$.Prove that the tangents to circumcirle of $\triangle BB^\prime I$ at $B^\prime$,$I$ intersect on $AC$. (A. Kuznetsov)

Convex quadrilateral $ABCD$ is given. Lines $BC$ and $AD$ intersect at point $O$, with $B$ lying on the segment $OC$, and $A$ on the segment $OD$. $I$ is the center of the circle inscribed in the $OAB$ triangle, $J$ is the center of the circle exscribed in the triangle $OCD$ touching the side of $CD$ and the extensions of the other two sides. The perpendicular from the midpoint of the segment $IJ$ on the lines $BC$ and $AD$ intersect the corresponding sides of the quadrilateral (not the extension) at points $X$ and $Y$. Prove that the segment $XY$ divides the perimeter of the quadrilateral$ABCD$ in half, and from all segments with this property and ends on $BC$ and $AD$, segment $XY$ has the smallest length.

A circle $k$ of radius $r$ is inscribed in $\vartriangle ABC$, tangent to the circle $k$, which are parallel respectively to the sides $AB, BC$ and $CA$ intersect the other sides of $\vartriangle ABC$ at points $M, N; P, Q$ and $L, T$ ($P, T \in AB$, $L, N \in BC$ and $M, Q\in AC$). Denote by $r_1,r_2,r_3$ the radii of inscribed circles in triangles $MNC, PQA$ and $LTB$. Prove that $r_1+r_2+r_3=r$.

Let $ABCD$ be a cyclic quadrilateral for which $|AD| =|BD|$. Let $M$ be the intersection of $AC$ and $BD$. Let $I$ be the incentre of $\triangle BCM$. Let $N$ be the second intersection pointof $AC$ and the circumscribed circle of $\triangle BMI$. Prove that $|AN| \cdot |NC| = |CD | \cdot |BN|$.

A paper square was bent by a line in such way that one vertex came to a side not containing this vertex. Three circles are inscribed into three obtained triangles (see Figure). Prove that one of their radii is equal to the sum of the two remaining ones. (L.Steingarts)

Let $M$ and $N$ be the midpoints of segments $BC$ and $AC$ of a triangle $ABC$, respectively. Let $Q$ be a point on the line through $N$ parallel to $BC$ such that $Q$ and $C$ are on opposite sides of $AB$ and $\vert QN\vert \cdot \vert BC\vert=\vert AB\vert \cdot \vert AC\vert$. Suppose that the circumcircle of triangle $AQN$ intersects the segment $MN$ a second time in a point $T \ne N$. Prove that there is a circle through points $T$ and $N$ touching both the side $BC$ and the incircle of triangle $ABC$.

Two triangles $ABC$ and $DEF$ have the same incircle. If a circle passes through $A,B,C,D,E$ prove that it also passes through $F$.

(a) A circle is inscribed in a triangle. The tangent points are the vertices of a second triangle in which another circle is inscribed. Its tangency points are the vertices of a third triangle. The angles of this triangle are identical to those of the first triangle. Find these angles. (b) A circle is inscribed in a scalene triangle. The tangent points are vertices of another triangle, in which a circle is inscribed whose tangent points are vertices of a third triangle, in which a third circle is inscribed, etc. Prove that the resulting sequence does not contain a pair of similar triangles.

Let $ABC$ be a triangle whose inscribed circle is $\omega$. Let $r_1$ be the line parallel to $BC$ and tangent to $\omega$, with $r_1 \ne BC$ and let $r_2$ be the line parallel to $AB$ and tangent to $\omega$ with $r_2 \ne AB$. Suppose that the intersection point of $r_1$ and $r_2$ lies on the circumscribed circle of triangle $ABC$. Prove that the sidelengths of triangle $ABC$ form an arithmetic progression.

Given a convex quadrilateral $ABCD$. We denote $I_A,I_B, I_C$ and $I_D$ centers of $\omega_A, \omega_B,\omega_C $and $\omega_D$,inscribed In the triangles $DAB, ABC, BCD$ and $CDA$, respectively.It turned out that $\angle BI_AA + \angle I_CI_AI_D = 180^\circ$. Prove that $\angle BI_BA + \angle I_CI_BI_D = 180^{\circ}$. (A. Kuznetsov)

The inscribed circle $\omega$ of the triangle $ABC$ touches its sides $BC, CA$ and $AB$ at points $A_1, B_1$ and $C_1$, respectively. Let $S$ be the intersection point of lines passing through points $B$ and $C$ and parallel to $A_1C_1$ and $A_1B_1$ respectively, $A_0$ be the foot of the perpendicular drawn from point $A_1$ on $B_1C_1$, $G_1$ be the centroid of triangle $A_1B_1C_1$, $P$ be the intersection point of the ray $G_1A_0$ with $\omega$. Prove that points $S, A_1$, and $P$ lie on a straight line.

In triangle $ABC$ ($AB\not= AC$), the incircle is tangent to the sides of $BC$ ,$CA$ , $AB$ at $D$ ,$E$, $F$ respectively. Let $AD$ meet the incircle again at point $P$, let $EF$ and the line passing through the point $P$ and perpendicular to $AD$ intersect at $Q$. Let $AQ$ intersect $DE$ at $X$ and $DF$ at $Y$. Prove that $AX=AY$. [i](tatari/nightmare)[/i]

Let $\Omega$ be the inscribed circle of a triangle $\vartriangle ABC$. Let $D, E$ and $F$ be the tangency points of $\Omega$ and the sides $BC, CA$ and $AB$, respectively, and let $AD, BE$ and $CF$ intersect $\Omega$ at $K, L$ and $M$, respectively, such that $D, E, F, K, L$ and $M$ are all distinct. The tangent line of $\Omega$ at $K$ intersects $EF$ at $X$, the tangent line of $\Omega$ at $L$ intersects $DE$ at $Y$ , and the tangent line of $\Omega$ at M intersects $DF$ at $Z$. Prove that $X,Y$ and $Z$ are collinear.

Let $ABCD$ be a square with incircle $\Gamma$. Let $M$ be the midpoint of the segment $[CD]$. Let $P \neq B$ be a point on the segment $[AB]$. Let $E \neq M$ be the point on $\Gamma$ such that $(DP)$ and $(EM)$ are parallel. The lines $(CP)$ and $(AD)$ meet each other at $F$. Prove that the line $(EF)$ is tangent to $\Gamma$

A triangle $ABC$ is given, in which the segment $BC$ touches the incircle and the corresponding excircle in points $M$ and $N$. If $\angle BAC = 2 \angle MAN$, show that $BC = 2MN$. (N.Beluhov)

Let $ABC$ be a triangle inscribed in the circle $K$ and consider a point $M$ on the arc $BC$ that do not contain $A$. The tangents from $M$ to the incircle of $ABC$ intersect the circle $K$ at the points $N$ and $P$. Prove that if $\angle BAC = \angle NMP$, then triangles $ABC$ and $MNP$ are congruent. Valentin Vornicu [hide= about Romania JBMO TST 2004 in aops]I found the Romania JBMO TST 2004 links [url=https://artofproblemsolving.com/community/c6h5462p17656]here [/url] but they were inactive. So I am asking for solution for the only geo I couldn't find using search. The problems were found [url=https://artofproblemsolving.com/community/c6h5135p16284]here[/url].[/hide]

Let $K,L,M$ be points on sides $AB,BC,CA$, respectively, of a triangle $ABC$ such that $AK/AB = BL/BC = CM/CA = 1/3$. Show that if the circumcircles of the triangles $AKM, BLK, CML$ are equal, then so are the incircles of these triangles.

Given is triangle $ABC$ with an inscribed circle with center $M$ and radius $r$. The tangent to this circle parallel to $BC$ intersects $AC$ in $D$ and $AB$ in $E$. The tangent to this circle parallel to $AC$ intersects $AB$ in $F$ and $BC$ in $G$. The tangent to this circle parallel to $AB$ intersects $BC$ in $H$ and $AC$ in $K$. Name the centers of the inscribed circles of triangle $AED$, triangle $FBG$ and triangle $KHC$ successively $M_A, M_B, M_C$ and the rays successively $r_A, r_B$ and $r_C$. Prove that $r_A + r_B + r_C = r$.

Given an equilateral triangle $ABC$ and a straight line $\ell$, passing through its center. Intersection points of this line with sides $AB$ and $BC$ are reflected wrt to the midpoints of these sides respectively. Prove that the line passing through the resulting points, touches the inscribed circle triangle $ABC$.

Let $ABC$ be a right-angled triangle, $I$ its incenter and $B_0, A_0$ points of tangency of the incircle with the legs $AC$ and $BC$ respectively. Let the perpendicular dropped to $AI$ from $A_0$ and the perpendicular dropped to $BI$ from $B_0$ meet at point $P$. Prove that the lines $CP$ and $AB$ are perpendicular.

$ABC$ triangle with $|AB|<|BC|<|CA|$ has the incenter $I$. The orthocenters of triangles $IBC, IAC$ and $IAB$ are $H_A, H_A$ and $H_A$. $H_BH_C$ intersect $BC$ at $K_A$ and perpendicular line from $I$ to $H_BH_B$ intersect $BC$ at $L_A$. $K_B, L_B, K_C, L_C$ are defined similarly. Prove that $$|K_AL_A|=|K_BL_B|+|K_CL_C|$$

Let $C_1$ and $C_2$ be the incircle and the circumcircle of the triangle $ABC$, respectively. Prove that, for any point $A'$ on $C_2$, there exist points $B'$ and $C'$ such that $C_1$ and $C_2$ are the incircle and the circumcircle of triangle $A'B'C'$, respectively.

A non-equilateral triangle $ABC$ is inscribed in a circle $\Gamma$ with centre $O$, radius $R$ and its incircle has centre $I$ and touches $BC,CA,AB$ at $D,E, F$, respectively. A circle with centre $I$ and radius $r$ intersects the rays $[ID), [IE), [IF)$ at $A',B',C'$. Show that the orthocentre $K$ of $\vartriangle A'B'C'$ is on the line $OI$ and that $\frac{IK}{IO}=\frac{r}{R}$ Michel Bataille .

The incircle of $\vartriangle ABC$ touches the sides $BC,CA,AB$ at $D,E,F$ respectively. A circle through $A$ and $B$ encloses $\vartriangle ABC$ and intersects the line $DE$ at points $P$ and $Q$. Prove that the midpoint of $AB$ lies on the circumircle of $\vartriangle PQF$.

Given triangle $ABC$, midpoint $M$ of the side $[BC]$, the centre $O$ of the inscribed circle. The line $(MO)$ crosses the height $AH$ in the point $E$. Prove that the distance $|AE|$ equals the inscribed circle radius.