Let $ABCD$ be a non-cyclic convex quadrilateral. The feet of perpendiculars from $A$ to $BC,BD,CD$ are $P,Q,R$ respectively, where $P,Q$ lie on segments $BC,BD$ and $R$ lies on $CD$ extended. The feet of perpendiculars from $D$ to $AC,BC,AB$ are $X,Y,Z$ respectively, where $X,Y$ lie on segments $AC,BC$ and $Z$ lies on $BA$ extended. Let the orthocenter of $\triangle ABD$ be $H$. Prove that the common chord of circumcircles of $\triangle PQR$ and $\triangle XYZ$ bisects $BH$.

Let $ABC$ be a an acute triangle. Points $A_1, B_1$ and $C_1$ are respectively the projections of the vertices $A, B$ and $C$ on the opposite sides of the triangle, the point $H$ is the orthocenter of the triangle, and the point $P$ is the middle of the segment $[AH]$. The lines $BH$ and $A_1C_1$, $P B_1$ and $AB$ intersect respectively at the points $M$ and $N$. Prove that the lines $MN$ and $BC$ are perpendicular.

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]

Given an acute $\vartriangle ABC$ whose orthocenter is denoted by $H$. A line $\ell$ passes $H$ and intersects $AB,AC$ at $P ,Q$ such that $H$ is the mid-point of $P,Q$. Assume the other intersection of the circumcircle of $\vartriangle ABC$ with the circumcircle of $\vartriangle APQ$ is $X$. Let $C'$ is the symmetric point of $C$ with respect to $X$ and $Y$ is the another intersection of the circumcircle of $\vartriangle ABC$ and $AO$, where O is the circumcenter of $\vartriangle APQ$. Show that $CY$ is tangent to circumcircle of $\vartriangle BCC'$. [img]https://1.bp.blogspot.com/-itG6m1ipAfE/XndLDUtSf7I/AAAAAAAALfc/iZahX6yNItItRSXkDYNofR5hKApyFH84gCK4BGAYYCw/s1600/2018%2Bimoc%2Bg3.png[/img]

In $\vartriangle ABC$ on the sides $BC, CA, AB$ mark feet of altitudes $H_1, H_2, H_3$ and the midpoint of sides $M_1, M_3, M_3$. Let $H$ be orthocenter $\vartriangle ABC$. Suppose that $X_2, X_3$ are points symmetric to $H_1$ wrt $BH_2$ and $CH_3$. Lines $M_3X_2$ and $M_2X_3$ intersect at point $X$. Similarly, $Y_3,Y_1$ are points symmetric to $H_2$ wrt $C_3H$ and $AH_1$.Lines $M_1Y_3$ and $M_3Y_1$ intersect at point $Y.$ Finally, $Z_1,Z_2$ are points symmetric to $H_3$ wrt $AH_1$ and $BH_2$. Lines $M_1Z_2$ and $M_2Z_1$ intersect at the point $Z$ Prove that $H$ is the incenter $\vartriangle XYZ$ .

A chord $AB$ is drawn in the circle, on which the point $P$ is selected in such a way that $AP = 2PB$. The chord $DE$ is perpendicular to the chord $AB $ and passes through the point $P$. Prove that the midpoint of the segment $AP$ is the orthocener of the triangle $AED$.

Given that $ABC$ is a triangle where $AB < AC$. On the half-lines $BA$ and $CA$ we take points $F$ and $E$ respectively such that $BF = CE = BC$. Let $M,N$ and $H$ be the mid-points of the segments $BF,CE$ and $BC$ respectively and $K$ and $O$ be the circumcenters of the triangles $ABC$ and $MNH$ respectively. We assume that $OK$ cuts $BE$ and $HN$ at the points $A_1$ and $B_1$ respectively and that $C_1$ is the point of intersection of $HN$ and $FE$. If the parallel line from $A_1$ to $OC_1$ cuts the line $FE$ at $D$ and the perpendicular from $A_1$ to the line $DB_1$ cuts $FE$ at the point $M_1$, prove that $E$ is the orthocenter of the triangle $A_1OM_1$.

Let $ABC$ be an acute-angled and non-isosceles triangle with orthocenter $H$. Let $O$ be the center of the circumscribed circle of triangle $ABC$ and let $K$ be center of the circumscribed circle of triangle $AHO$. Prove that the reflection of $K$ wrt $OH$ lies on $BC$.

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]

We are given an acute triangle $ABC$ with $AB > AC$ and orthocenter $H$. The point $E$ lies symmetric to $C$ with respect to the altitude $AH$. Let $F$ be the intersection of the lines $EH$ and $AC$. Prove that the circumcenter of the triangle $AEF$ lies on the line $AB$. (Karl Czakler)

Let $ABCD$ be a rhombus, let $APQC$ be a parallelogram such that the point $B$ lies inside it and the side $AP$ is equal to the side of the rhombus. Prove that $B$ is the orthocenter of the triangle $DPQ$. Egor Bakaev

The measure of the angle $\angle A$ of the acute triangle $ABC$ is $60^o$, and $HI = HB$, where $I$ and $H$ are the incenter and the orthocenter of the triangle $ABC$. Find the measure of the angle $\angle B$.

In a parallelogram $ABCD$, the perpendiculars from $A$ to $BC$ and $CD$ meet the line segments $BC$ and $CD$ at the points $E$ and $F$ respectively. Suppose $AC = 37$ cm and $EF = 35$ cm. Let $H$ be the orthocentre of $\vartriangle AEF$. Find the length of $AH$ in cm. Show the steps in your calculations.

Let $ABC$ be an acute and scalene of circumcircle $\Gamma$ and orthocenter $H$. Let $A_1,B_1,C_1$ be the second intersection points of the lines $AH, BH, CH$ with $\Gamma$, respectively. The lines that pass through $A_1,B_1,C_1$ and are parallel to $BC,CA, AB$ intersect again to $\Gamma$ at $A_2,B_2,C_2$, respectively. Let $M$ be the intersection point of $AC_2$ and $BC_1, N$ the intersection point of $BA_2$ and $CA_1$, and $P$ the intersection point of $CB_2$ and $AB_1$. Prove that $\angle MNB = \angle AMP$ .

Let $ABC$ be an acutangle triangle and $O, H$ its circumcenter, orthocenter, respectively. If $\frac{AB}{\sqrt2}=BH=OB$, calculate the angles of the triangle $ABC$ .

Let $ABC$ be an acute-angled triangle, $O$ its circumcenter and $H$ its orthocenter. The orthogonal projection of the vertex $A$ to the line $BC$ lies on the perpendicular bisector of the segment $AC$. Compute $\frac{CH}{BO}$ . (J. Willemson)

Consider a triangle $ABC$ with $AB < AC$ and let $H$ and $O$ be its orthocenter and circumcenter, respectively. A line starting from $B$ cuts the lines $AO$ and $AH$ at $M$ and $M'$ so that $M'$ is the midpoint of $BM$. Another line starting from $C$ cuts the lines $AH$ and $AO$ at $N$ and $N'$ so that $N'$ is the midpoint of $CN$. Prove that $M, M', N, N'$ are on the same circle.

Determine the locus of vertices of triangles which have prescribed orthocenter and center of circumcircle.

The orthocenter of an acute triangle $ABC$ is $H$. Let $K$ and $P$ be the midpoints of lines $BC$ and $AH$, respectively. The angle bisector drawn from the vertex $A$ of the triangle $ABC$ intersects with line $KP$ at $D$. Prove that $HD\perp AD$.

The altitudes of the acute-angled triangle $ABC$ intersect at the point $H$. On the segments $BH$ and $CH$, the points $B_1$ and $C_1$ are marked, respectively, so that $B_1C_1 \parallel BC$. It turned out that the center of the circle $\omega$ circumscribed around the triangle $B_1HC_1$ lies on the line $BC$. Prove that the circle $\Gamma$, which is circumscribed around the triangle $ABC$, is tangent to the circle $\omega$ .

Let $H$ be an orhocenter of an acute triangle $ABC$ and $M$ midpoint of side $BC$. If $D$ and $E$ are foots of perpendicular of $H$ on internal and external angle bisector of angle $\angle BAC$, prove that $M$, $D$ and $E$ are collinear

Let $H$ and $O$ be the orthocenter and the circumcenter of the triangle $ABC$. Line $OH$ intersects the sides $AB, AC$ at points $X, Y$ correspondingly, so that $H$ belongs to the segment $OX$. It turned out that $XH = HO = OY$. Find $\angle BAC$. [i](Proposed by Oleksii Masalitin)[/i]

Given an acute triangle $ABC$, $(c)$ its circumcircle with center $O$ and $H$ the orthocenter of the triangle $ABC$. The line $AO$ intersects $(c)$ at the point $D$. Let $D_1, D_2$ and $H_2, H_3$ be the symmetrical points of the points $D$ and $H$ with respect to the lines $AB, AC$ respectively. Let $(c_1)$ be the circumcircle of the triangle $AD_1D_2$. Suppose that the line $AH$ intersects again $(c_1)$ at the point $U$, the line $H_2H_3$ intersects the segment $D_1D_2$ at the point $K_1$ and the line $DH_3$ intersects the segment $UD_2$ at the point $L_1$. Prove that one of the intersection points of the circumcircles of the triangles $D_1K_1H_2$ and $UDL_1$ lies on the line $K_1L_1$.

The vertices of the pentagon $ABCDE$ are on a circle, and the points $H_1, H_2, H_3,H_4$ are the orthocenters of the triangles $ABC, ABE, ACD, ADE$ respectively . Prove that the quadrilateral determined by the four orthocenters is square if and only if $BE \parallel CD$ and the distance between them is $\frac{BE + CD}{2}$.

Let $O$ be the circumcenter and $H$ the orthocenter of an acute-triangle $ABC$. The perpendicular bisector of $AO$ intersects the line $BC$ at point $S$. Let $L$ be the midpoint of $OH$. Prove that $\angle OAH = \angle LSA$.