Found problems: 63
Let $ABC$ be a triangle with bisectors $BE$ and $CF$ meet at $I$. Let $D$ be the projection of $I$ on the $BC$. Let M and $N$ be the orthocenters of triangles $AIF$ and $AIE$, respectively. Lines $EM$ and $FN$ meet at $P.$ Let $X$ be the midpoint of $BC$. Let $Y$ be the point lying on the line $AD$ such that $XY \perp IP$. Prove that line $AI$ bisects the segment $XY$.
[i]Proposed by Tran Quang Hung - Vietnam[/i]
Let $ABCDE$ be a convex pentagon such that angles $CAB$, $BCA$, $ECD$, $DEC$ and $AEC$ are equal. Prove that $CE$ bisects $BD$.
We are given a right triangle $ABC$ and the median $BD$ drawn from the vertex $B$ of the right angle. Let the circle inscribed in $\vartriangle ABD$ be tangent to side $AD$ at $K$. Find the angles of $\vartriangle ABC$ if $K$ divides $AD$ in halves.
Divide a segment in halves using a right triangle. (With a right triangle one can draw straight lines and erect perpendiculars but cannot draw perpendiculars.)
$AM$ is the median of triangle $ABC$. A perpendicular $CC_1$ is drawn from point $C$ on the bisector of angle $\angle CMA$, a perpendicular $BB_1$ is drawn from point $B$ on the bisector of angle $\angle BMA$. Prove that line $AM$ intersects segment $B_1C_1$ at its midpoint.
The incircle of triangle $ABC$ touches its sides at points $A'$, $B'$, $C'$. $I$ is its center. Straight line $B'I$ intersects segment $A'C'$ at point $P$. Prove that straight line $BP$ passes through the midpoint of $AC$.
Let $OP$ and $OQ$ be the perpendiculars from the circumcenter $O$ of a triangle $ABC$ to the internal and external bisectors of the angle $B$. Prove that the line$ PQ$ divides the segment connecting midpoints of $CB$ and $AB$ into two equal parts.
(Artemiy Sokolov)
In triangle $ABC$, points $A_1,B_1,C_1$ are bases of altitudes from vertices $A,B,C$, and points $C_A,C_B$ are the projections of $C_1$ to $AC$ and $BC$ respectively. Prove that line $C_AC_B$ bisects the segments $C_1A_1$ and $C_1B_1$.
Let $ABC$ be an acute, non-isosceles triangle inscribed in (O) and $BB'$, $CC'$ are altitudes. Denote $E, F$ as the intersections of $BB'$, $CC'$ with $(O)$ and $D, P, Q$ are projections of $A$ on $BC$, $CE$, $BF$. Prove that the perpendicular bisectors of $PQ$ bisects two segments $AO$, $BC$.
Let $H$ be the orthocentre of an acute triangle $ABC$ with $BC > AC$, inscribed in a circle $\Gamma$. The circle with centre $C$ and radius $CB$ intersects $\Gamma$ at the point $D$, which is on the arc $AB$ not containing $C$. The circle with centre $C$ and radius $CA$ intersects the segment $CD$ at the point $K$. The line parallel to $BD$ through $K$, intersects $AB$ at point $L$. If $M$ is the midpoint of $AB$ and $N$ is the foot of the perpendicular from $H$ to $CL$, prove that the line $MN$ bisects the segment $CH$.
Let $ABC$ be a triangle with $AB \ne AC$ and incenter $I$. The incircle touches $BC$ at $D$. Let $M$ be the midpoint of $BC$ . Show that the line $IM$ bisects segment $AD$ .
On the side $AB$ of the cyclic quadrilateral $ABCD$ there is a point $X$ such that diagonal $AC$ bisects the segment $DX$, and the diagonal $BD$ bisects the segment $CX$. What is the smallest possible ratio $|AB | : |CD|$ in such a quadrilateral ?
Let $CH$ be the altitude of triangle ABC, $O$ be the center of the circle circumscribed around it. Point $T$ is the projection of point $C$ on the line $TO$. Prove that the line $TH$ bisects the side $BC$.
Given a pointed triangle $ABC$ with orthocenter $H$. Let $D$ be the point such that the quadrilateral $AHCD$ is parallelogram. Let $p$ be the perpendicular to the direction $AB$ through the midpoint $A_1$ of the side $BC$. Denote the intersection of the lines $p$ and $AB$ with $E$, and the midpoint of the length $A_1E$ with $F$. The point where the parallel to the line $BD$ through point $A$ intersects $p$ denote by $G$. Prove that the quadrilateral $AFA_1C$ is cyclic if and only if the lines $BF$ passes through the midpoint of the length $CG$.
Let $ABC$ be a triangle and $ABDE, BCFZ, CAKL$ be three similar rectangles constructed externally of the triangle. Let $A'$ be the intersection of $EF$ and $ZK, B'$ the intersection of $KZ$ and $DL$, and $C'$ the intersection of $DL$ and $EF$. Prove that $AA'$ passes through the midpoint of the line segment $B'C'$.
Kostas Vittas
Let $ABCDE$ be a convex pentagon such that $\angle BAC = \angle CAD = \angle DAE$ and $\angle ABC = \angle ACD = \angle ADE$. Diagonals $BD$ and $CE$ meet at $P$. Prove that $AP$ bisects side $CD$.
Let $A$, $B$, $C$ and $D$ be points on the same circumference with $\angle BCD=90^\circ$. Let $P$ and $Q$ be the projections of $A$ onto $BD$ and $CD$, respectively. Prove that $PQ$ cuts the segment $AC$ into equal parts.
$PQ, CD$ are parallel chords of a circle. The tangent at $D$ cuts $PQ$ at $T$ and $B$ is the point of contact of the other tangent from $T$ (Fig. ). Prove that $BC$ bisects $PQ$.
[img]https://cdn.artofproblemsolving.com/attachments/2/f/22f69c03601fbb8e388e319cd93567246b705c.png[/img]
Let $\vartriangle ABC$ be an acute triangle with $AB > AC$. Let $P$ be the foot of the altitude from $C$ to $AB$ and let $Q$ be the foot of the altitude from $B$ to $AC$. Let $X$ be the intersection of $PQ$ and $BC$. Let the intersection of the circumcircles of triangle $\vartriangle AXC$ and triangle $\vartriangle PQC$ be distinct points: $C$ and $Y$ . Prove that $PY$ bisects $AX$.
The altitudes $AN$ and $BM$ are drawn in triangle $ABC$. Prove that the perpendicular bisector to the segment $NM$ divides the segment $AB$ in half.
From a point $P$, two tangent lines are drawn to a circle $\Gamma$, which touch it at points $A$ and $B$. A circle $\Phi$ is drawn with center at $P$ and passes through $A$ and $B$ and is taken a point $R$ that is on the circumference $\Phi$ and in the interior of $\Gamma$. The straight line $PR$ intersects $\Gamma$ at the points $S$ and $Q$. The straight lines $AR$ and $BR$ meet $\Gamma$ again at points $C$ and $D$, respectively. Prove that $CD$ passes through the midpoint of $SQ$.
Given acute triangle $\vartriangle ABC$ with orthocenter $H$ and circumcenter $O$ ($O \ne H$) . Let $\Gamma$ be the circumcircle of $\vartriangle BOC$ . Segment $OH$ untersects $\Gamma$ at point $P$. Extension of $AO$ intersects $\Gamma$ at point $K$. If $AP \perp OH$, prove that $PK$ bisects $BC$.
[img]https://cdn.artofproblemsolving.com/attachments/a/b/267053569c41692f47d8f4faf2a31ebb4f4efd.png[/img]
The incircle of triangle $ABC$ centered at $I$ touches $CA,AB$ at points $E,F$ respectively. Let points $M,N$ of line $EF$ be such that $CM=CE$ and $BN=BF$. Lines $BM$ and $CN$ meet at point $P$. Prove that $PI$ bisects segment $MN$.
The segment $AB$ is the diameter of the circle. The points $M$ and $C$ belong to this circle and are located in different half-planes relative to the line $AB$. From the point $M$ the perpendiculars $MN$ and $MK$ are drawn on the lines $AB$ and $AC$, respectively. Prove that the line $KN$ intersects the segment $CM$ in its midpoint.
(Igor Nagel)
Let $H$ be the orthocenter of an acute-angled triangle $ABC$ and $P$ a point on the circumcenter of triangle $ABC$. Prove that the Simson line of $P$ bisects the segment $[P H]$.