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 a triangle, $O{}$ be its circumcenter, $I{}$ its incenter and $I_A,I_B,I_C$ the excenters. Let $M$ be the midpoint of $BC$ and $H_1$ and $H_2$ be the orthocenters of the triangles $MII_A$ and $MI_BI_C$. Prove that the parallel to $BC$ through $O$ passes through the midpoint of the segment $H_1H_2$. [i]Proposed by David Anghel[/i]

The median $AM$ is drawn in the acute-angled triangle $ABC$ with different sides. Its extension intersects the circumscribed circle $w$ of this triangle at the point $P$. Let $A {{H} _ {1}}$ be the altitude $\Delta ABC$, $H$ be the point of intersection of its altitudes. The rays $MH$ and $P {{H} _ {1}}$ intersect the circle $w$ at the points $K$ and $T$, respectively. Prove that the circumscribed circle of $\Delta KT {{H} _ {1}}$ touches the segment $BC$. (Hilko Danilo)

Let $ABC$ be an acute triangle , with $AB \neq AC$ and denote its orthocenter by $H$ . The point $D$ is located on the side $BC$ and the circumcircles of the triangles $ABD$ and $ACD$ intersects for the second time the lines $AC$ , respectively $AB$ in the points $E$ respectively $F$. If we denote by $P$ the intersection point of $BE$ and $CF$ then show that $HP \parallel BC$ if and only if $AD$ passes through the circumcenter of the triangle $ABC$.

Let $\omega$ be the circumcircle of triangle $ABC$. A point $B_1$ is chosen on the prolongation of side $AB$ beyond point B so that $AB_1 = AC$. The angle bisector of $\angle BAC$ meets $\omega$ again at point $W$. Prove that the orthocenter of triangle $AWB_1$ lies on $\omega$ . (A.Tumanyan)

Let $ABC$ be a scalene acute triangle whose orthocenter is $H$. $M$ is the midpoint of segment $BC$. $N$ is the point where the segment $AM$ intersects the circle determined by $B, C$, and $H$. Show that lines $HN$ and $AM$ are perpendicular.

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.

In a quadrilateral $ABCD$ the intersection of the diagonals is called $P$. Point $X$ is the orthocentre of triangle $PAB$. (The orthocentre of a triangle is the point where the three altitudes of the triangle intersect.) Point $Y$ is the orthocentre of triangle $PCD$. Suppose that $X$ lies inside triangle $PAB$ and $Y$ lies inside triangle $PCD$. Moreover, suppose that $P$ is the midpoint of line segment $XY$ . Prove that $ABCD$ is a parallelogram. [asy] import geometry; unitsize (1.5 cm); pair A, B, C, D, P, X, Y; A = (0,0); B = (2,-0.5); C = (3.5,2.2); D = A + C - B; P = (A + C)/2; X = orthocentercenter(A,B,P); Y = orthocentercenter(C,D,P); draw(A--B--C--D--cycle); draw(A--C); draw(B--D); draw(A--extension(A,X,B,P), dotted); draw(B--extension(B,X,A,P), dotted); draw(P--extension(P,X,A,B), dotted); draw(C--extension(C,Y,D,P), dotted); draw(D--extension(D,Y,C,P), dotted); draw(P--extension(P,Y,C,D), dotted); dot("$A$", A, W); dot("$B$", B, S); dot("$C$", C, E); dot("$D$", D, N); dot("$P$", P, E); dot("$X$", X, NW); dot("$Y$", Y, SE); [/asy]

The circumcenter and orthocenter of $ABC$ are $O$ and $H$, respectively. Let $XACH$ be a parallelogram. Show that if $OH$ is parallel to $BC$, then $OX$ and $AB$ intersect at some point on the perpendicular bisector of $AH$. [i]proposed by USJL[/i]

Let $ABC$ be a triangle in which $AB < AC, D$ is the foot of the altitude from $A, H$ is the orthocenter, $O$ is the circumcenter, $M$ is the midpoint of the side $BC, A'$ is the reflection of $A$ across $O$, and $S$ is the intersection of the tangents at $B$ and $C$ to the circumcircle. The tangent at $A'$ to the circumcircle intersects $SC$ and $SB$ at $X$ and $Y$ , respectively. If $M,S,X,Y$ are concyclic, prove that lines $OD$ and $SA'$ are parallel.

Let $AA_1, BB_1, CC_1$ be the heights of triangle $ABC$ and $H$ be its orthocenter. Liune $\ell$ parallel to $AC$, intersects straight lines $AA_1$ and $CC_1$ at points $A_2$ and $C_2$, respectively. Suppose that point $B_1$ lies outside the circumscribed circle of triangle $A_2 HC_2$. Let $B_1P$ and $B_1T$ be tangent to of this circle. Prove that points $A_1, C_1, P$, and $T$ are cyclic.

Constant points $B$ and $C$ lie on the circle $\omega$. The point middle of $BC$ is named $M$ by us. Assume that $A$ is a variable point on the $\omega$ and $H$ is the orthocenter of the triangle $ABC$. From the point $H$ we drop a perpendicular line to $MH$ to intersect the lines $AB$ and $AC$ at $X$ and $Y$ respectively. Prove that with the movement of $A$ on the $\omega$, the orthocenter of the triangle $AXY$ also moves on a circle.

Given a triangle $ ABC$. Let $ G$, $ I$, $ H$ be the centroid, the incenter and the orthocenter of triangle $ ABC$, respectively. Prove that $ \angle GIH > 90^{\circ}$.

Given a scalene triangle $\vartriangle ABC$ with orthocenter $H$. The midpoint of $BC$ is denoted by $M$. $AH$ intersects the circumcircle at $D \ne A$ and $DM$ intersects circumcircle of $\vartriangle ABC$ at $T\ne D$. Now, assume the reflection points of $M$ with respect to $AB,AC,AH$ are $F,E,S$. Show that the midpoints of $BE,CF,AM,TS$ are concyclic. [img]https://3.bp.blogspot.com/-v7D_A66nlD0/XnYNJussW9I/AAAAAAAALeQ/q6DMQ7w6QtI5vLwBcKqp4010c3XTCj3BgCK4BGAYYCw/s1600/imoc2019g2.png[/img]

Let $ABC$ be an acute triangle, and its altitudes $AX,BY,CZ$ concurrent at $H$. Construct circles $(K_a), (K_b), (K_c)$ circumscribing the triangles $AY Z, BZX, CXY$ . Construct a circle $(K)$ that is internally tangent to all the three circles $(Ka), (K_b), (K_c)$. Prove that $(K)$ is tangent to the circumcircle $(O)$ of the triangle $ABC$. Đỗ Thanh Sơn

Altitudes $AD$ and $BE$ of an acute triangle $ABC$ intersect at $H$. Let $P \ne E$ be the point of tangency of the circle with radius $HE$ centred at $H$ with its tangent line going through point $C$, and let $Q \ne E$ be the point of tangency of the circle with radius $BE$ centred at $B$ with its tangent line going through $C$. Prove that the points $D, P$ and $Q$ are collinear.

Suppose that points $X, Y,Z$ are located on sides $BC, CA$, and $AB$, respectively, of triangle $ABC$ in such a way that triangle $XY Z$ is similar to triangle $ABC$. Prove that the orthocenter of triangle $XY Z$ is the circumcenter of triangle $ABC.$

Let $ABC$ be a scalene triangle, $\Gamma$ its circumscribed circle and $H$ the point where the altitudes of triangle $ABC$ meet. The circumference with center at $H$ passing through $A$ cuts $\Gamma$ at a second point $D$. In the same way, the circles with center at $H$ and passing through $B$ and $C$ cut $\Gamma$ again at points $E$ and $F$, respectively. Prove that $H$ is also the point in which the altitudes of the triangle $DEF$ meet.

$ABC$ is a triangle, $H$ its orthocenter, $I$ its incenter, $O$ its circumcenter and $\omega$ its circumcircle. Line $CI$ intersects circle $\omega$ at point $D$ different from $C$. Assume that $AB = ID$ and $AH = OH$. Find the angles of triangle $ABC$.

In a circle of center $O$ and radius $r$, a triangle $ABC$ of orthocenter $H$ is inscribed. It is considered a triangle $A'B'C'$ whose sides have by length the measurements of the segments $AB, CH$ and $2r$. Determine the triangle $ABC$ so that the area of the triangle $A'B'C'$ is maximum.

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

The external angle bisector of the angle $A$ of an acute-angled triangle $ABC$ meets the circumcircle of $\vartriangle ABC$ at point $T$. The perpendicular from the orthocenter $H$ of $\vartriangle ABC$ to the line $TA$ meets the line $BC$ at point $P$. The line $TP$ meets the circumcircce of $\vartriangle ABC$ at point $D$. Prove that $AB^2+DC^2=AC^2+BD^2$ A. Voidelevich

Altitudes $AD$ and $BE$ of an acute triangle $ABC$ intersect at $H$. Let $C_1 (H,HE)$ and $C_2(B,BE)$ be two circles tangent at $AC$ at point $E$. Let $P\ne E$ be the second point of tangency of the circle $C_1 (H,HE)$ with its tangent line going through point $C$, and $Q\ne E$ be the second point of tangency of the circle $C_2(B,BE)$ with its tangent line going through point $C$. Prove that points $D, P$, and $Q$ are collinear.

The altitudes of triangle $ABC$ intersect at a point $H$.Find $\angle ACB$ if it is known that $AB = CH$.

We draw two lines $(\ell_1) , (\ell_2)$ through the orthocenter $H$ of the triangle $ABC$ such that each one is dividing the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle.