In the acute triangle $ABC$, with $AB \ne BC$, let $T$ denote the midpoint of the side $[AC], A_1$ and $C_1$ denote the feet of the altitudes drawn from $A$ and $C$, respectively. Let $Z$ be the intersection point of the tangents in $A$ and $C $ to the circumcircle of triangle $ABC, X$ be the intersection point of lines $ZA$ and $A_1C_1$ and $Y$ be the intersection point of lines $ZC$ and $A_1C_1$. a) Prove that $T$ is the incircle of triangle $XYZ$. b) The circumcircles of triangles $ABC$ and $A_1BC_1$ meet again at $D$. Prove that the orthocenter $H$ of triangle $ABC$ is on the line $TD$. c) Prove that the point $D$ lies on the circumcircle of triangle $XYZ$.

Let $ABC$ be a triangle inscribed in circle $(O),$ with its altitudes $BE, CF$ intersect at orthocenter $H$ ($E \in AC, F \in AB$). Let $M$ be the midpoint of $BC, K$ be the orthogonal projection of $H$ on $AM$. $EF$ intersects $BC$ at $P$. Let $Q$ be the intersection of tangent of $(O)$ which passes through $A$ with $BC, T$ be the reflection of $Q$ through $P$. Prove that $\angle OKT = 90^o$.

Altitudes $AA_1$ and $CC_1$ of an acute-angled triangle $ABC$ intersect at point $H$. A straight line passing through point $H$ parallel to line $A_1C_1$ intersects the circumscribed circles of triangles $AHC_1$ and $CHA_1$ at points $X$ and $Y$, respectively. Prove that points $X$ and $Y$ are equidistant from the midpoint of segment $BH$.

Let $ABC$ be a triangle inscribed in the circle $(O)$. The bisector of $\angle BAC$ cuts the circle $(O)$ again at $D$. Let $DE$ be the diameter of $(O)$. Let $G$ be a point on arc $AB$ which does not contain $C$. The lines $GD$ and $BC$ intersect at $F$. Let $H$ be a point on the line $AG$ such that $FH \parallel AE$. Prove that the circumcircle of triangle $HAB$ passes through the orthocenter of triangle $HAC$.

Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$ [i]Proposed by Michael Ren[/i]

Let $I_a$ be the $A$-excenter of a scalene triangle $ABC$. And let $M$ be the point symmetric to $I_a$ about line $BC$. Prove that line $AM$ is parallel to the line through the circumcenter and the orthocenter of triangle $I_aCB$.

Given an acute $\vartriangle ABC$ with orthocenter $H$. Let $M_a$ be the midpoint of $BC. M_aH$ intersects the circumcircle of $\vartriangle ABC$ at $X_a$ and $AX_a$ intersects $BC$ at $Y_a$. Define $Y_b, Y_c$ in a similar way. Prove that $Y_a, Y_b,Y_c$ are collinear. [img]https://2.bp.blogspot.com/-yjISBHtRa0s/XnSKTrhhczI/AAAAAAAALds/e_rvs9glp60L1DastlvT0pRFyP7GnJnCwCK4BGAYYCw/s320/imoc2017%2Bg4.png[/img]

Let $ABC$ be triangle with $H$ is the orthocenter and $I$ is incenter. Denote $A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2}$ be the points on the rays $AB, AC, BC, CA, CB$, respectively such that $$AA_{1} = AA_{2} = BC, BB_{1} = BB_{2} = CA, CC_{1} = CC_{2} = AB.$$ Suppose that $B_{1}B_{2}$ cuts $C_{1}C_{2}$ at $A'$, $C_{1}C_{2}$ cuts $A_{1}A_{2}$ at $B'$ and $A_{1}A_{2}$ cuts $B_{1}B_{2}$ at $C'$. a) Prove that area of triangle $A'B'C'$ is smaller than or equal to the area of triangle $ABC$. b) Let $J$ be circumcenter of triangle $A'B'C'$. $AJ$ cuts $BC$ at $R$, $BJ$ cuts $CA$ at $S$ and $CJ$ cuts $AB$ at $T$. Suppose that $(AST), (BTR), (CRS)$ intersect at $K$. Prove that if triangle $ABC$ is not isosceles then $HIJK$ is a parallelogram.

For any point $X$ inside an acute-angled triangle $ABC$ we define $$f(X)=\frac{AX}{A_1X}\cdot \frac{BX}{B_1X}\cdot \frac{CX}{C_1X}$$ where $A_1, B_1$, and $C_1$ are the intersection points of the lines $AX, BX,$ and $CX$ with the sides $BC, AC$, and $AB$, respectively. Let $H, I$, and $G$ be the orthocenter, the incenter, and the centroid of the triangle $ABC$, respectively. Prove that $f(H) \ge f(I) \ge f(G)$ . (D. Bazylev)

Given a scalene triangle $\vartriangle ABC$ has orthocenter $H$ and circumcircle $\Omega$. The tangent lines passing through $A,B,C$ are $\ell_a,\ell_b,\ell_c$. Suppose that the intersection of $\ell_b$ and $\ell_c$ is $D$. The foots of $H$ on $\ell_a,AD$ are $P,Q$ respectively. Prove that $PQ$ bisects segment $BC$. [img]https://4.bp.blogspot.com/-iiQoxMG8bEs/XnYNK7R8S3I/AAAAAAAALeY/FYvSuF6vQQsofASnXJUgKZ1T9oNnd-02ACK4BGAYYCw/s400/imoc2019g3.png[/img]

Find the angle $\angle A$ of a triangle $ABC$, when we know it's altitudes $BD$ and $CE$ intersect in an interior point $H$ of the triangle and $BH=2HD$ and $CH=HE$.

Let $ABC$ be an acute triangle with orthocenter $H$. Points $A_1$, $B_1$, $C_1$ are chosen in the interiors of sides $BC$, $CA$, $AB$, respectively, such that $\triangle A_1B_1C_1$ has orthocenter $H$. Define $A_2 = \overline{AH} \cap \overline{B_1C_1}$, $B_2 = \overline{BH} \cap \overline{C_1A_1}$, and $C_2 = \overline{CH} \cap \overline{A_1B_1}$. Prove that triangle $A_2B_2C_2$ has orthocenter $H$. [i]Ankan Bhattacharya[/i]

Let $H$ be the orthocenter of triangle $ABC$. The tangents to the circumcircles of triangles $CHB$ and $AHB$ at point $H$ meet $AC$ at points $A_1$ and $C_1$ respectively. Prove that $A_1H = C_1H$.

An acute triangle $ABC$ is given. Let $AD$ be its altitude, let $H$ and $O$ be its orthocenter and its circumcenter, respectively. Let $K$ be the point on the segment $AH$ with $AK=HD$; let $L$ be the point on the segment $CD$ with $CL=DB$. Prove that line $KL$ passes through $O$.

Let $A_1, B_1$ and $C_1$ be points on sides $BC$, $CA$ and $AB$ of an acute triangle $ABC$ respectively, such that $AA_1$, $BB_1$ and $CC_1$ are the internal angle bisectors of triangle $ABC$. Let $I$ be the incentre of triangle $ABC$, and $H$ be the orthocentre of triangle $A_1B_1C_1$. Show that $$AH + BH + CH \geq AI + BI + CI.$$

Let $ABC$ be a triangle with orthocenter $H$. Let $P$ be any point of the plane of the triangle. Let $\Omega$ be the circle with the diameter $AP$ . The circle $\Omega$ cuts $CA$ and $AB$ again at $E$ and $F$ , respectively. The line $PH$ cuts $\Omega$ again at $G$. The tangent lines to $\Omega$ at $E, F$ intersect at $T$. Let $M$ be the midpoint of $BC$ and $L$ be the point on $MG$ such that $AL$ and $MT$ are parallel. Prove that $LA$ and $LH$ are orthogonal. Lê Phúc Lữ

Let $ABC$ be an acute-angled triangle in which $\angle BAC = 60^o$ and $AB> AC$. Let $H$ and $I$ denote the points of intersection of the altitudes and angle bisectors of this triangle, respectively. Find the ratio $\angle ABC: \angle AHI$.

Let $ABC$ be an acute-angled triangle in which $AB <AC$. On the side $BC$ mark a point $D$ such that $AD = AB$, and on the side $AB$ mark a point $E$ such that the segment $DE$ passes through the orthocenter of triangle $ABC$. Prove that the center of the circumcircle of triangle $ADE$ lies on the segment $AC$.

Point $M$ is a midpoint of side $BC$ of a triangle $ABC$ and $H$ is the orthocenter of $ABC$. $MH$ intersects the $A$-angle bisector at $Q$. Points $X$ and $Y$ are the projections of $Q$ on sides $AB$ and $AC$. Prove that $XY$ passes through $H$.

Let $ABC$ be a triangle inscribed in circle $\Gamma$ with center $O$. Let $H$ be the orthocenter of triangle $ABC$ and let $K$ be the midpoint of $OH$. Tangent of $\Gamma$ at $B$ intersects the perpendicular bisector of $AC$ at $L$. Tangent of $\Gamma$ at $C$ intersects the perpendicular bisector of $AB$ at $M$. Prove that $AK$ and $LM$ are perpendicular. by Michael Sarantis, Greece

Let the points $M$ and $N$ in the triangle $ABC$ be the midpoints of the sides $BC$ and $AC$, respectively. It is known that the point of intersection of the altitudes of the triangle $ABC$ coincides with the point of intersection of the medians of the triangle $AMN$. Find the value of the angle $ABC$.

Let $ABC$ be an acute, non isosceles with $I$ is its incenter. Denote $D, E$ as tangent points of $(I)$ on $AB,AC$, respectively. The median segments respect to vertex $A$ of triangles $ABE$ and $ACD$ meet$ (I)$ at$ P,Q,$ respectively. Take points $M, N$ on the line $DE$ such that $AM \parallel BE$ and $AN \parallel C D$ respectively. a) Prove that $A$ lies on the radical axis of $(MIP)$ and $(NIQ)$. b) Suppose that the orthocenter $H$ of triangle $ABC$ lies on $(I)$. Prove that there exists a line which is tangent to three circles of center $A, B, C$ and all pass through $H$.

Let $H$ be the orthocenter of an acute triangle $ABC$. (The orthocenter is the point at the intersection of the three altitudes. An acute triangle has all angles less than $90^o$.) Draw three circles: one passing through $A, B$, and $H$, another passing through $B, C$, and $H$, and finally, one passing through $C, A$, and $H$. Prove that the triangle whose vertices are the centers of those three circles is congruent to triangle $ABC$.

Let $ABC$ be an acute-angled triangle with height intersection $H$. Let $G$ be the intersection of parallel of $AB$ through $H$ with the parallel of $AH$ through $B$. Let $I$ be the point on the line $GH$, so that $AC$ bisects segment $HI$. Let $J$ be the second intersection of $AC$ and the circumcircle of the triangle $CGI$. Show that $IJ = AH$

Given a circle and a fixed point $P$ not lying on it. Find the geometrical locus of the orthocenters of the triangles $ABP$, where $AB$ is the diameter of the circle.