Olympiad problems

2021 China Second Round A2, 1

As shown in the figure, in the acute angle $\vartriangle ABC$, $AB > AC$, $M$ is the midpoint of the minor arc $BC$ of the circumcircle $\Omega$ of $\vartriangle ABC$. $K$ is the intersection point of the bisector of the exterior angle $\angle BAC$ and the extension line of $BC$. From point $A$ draw a line perpendicular on $BC$ and take a point $D$ (different from $A$) on that line , such that $DM = AM$. Let the circumscribed circle of $\vartriangle ADK$ intersect the circle $\Omega$ at point $A$ and at another point $T$. Prove that $AT$ bisects line segment $BC$. [img]https://cdn.artofproblemsolving.com/attachments/1/3/6fde30405101620828d63ae31b8c0ffcec972f.png[/img]

Kyiv City MO Juniors 2003+ geometry, 2010.9.4

Tags: geometry , circumcircle , bisects segment

In an acute-angled triangle $ABC$, the point $O$ is the center of the circumcircle, $CH$ is the height of the triangle, and the point $T$ is the foot of the perpendicular dropped from the vertex $C$ on the line $AO$. Prove that the line $TH$ passes through the midpoint of the side $BC$ .

Swiss NMO - geometry, 2013.7

Tags: geometry , bisects segment , equal angles

Let $O$ be the center of the circle of the triangle $ABC$ with $AB \ne AC$. Furthermore, let $S$ and $T$ be points on the rays $AB$ and $AC$, such that $\angle ASO = \angle ACO$ and $\angle ATO = \angle ABO$. Show that $ST$ bisects the segment $BC$.

2012 China Northern MO, 7

Tags: bisects segment , pentagon , geometry

As shown in figure , in the pentagon $ABCDE$, $BC = DE$, $CD \parallel BE$, $AB>AE$. If $\angle BAC = \angle DAE$ and $\frac{AB}{BD}=\frac{AE}{ED}$. Prove that $AC$ bisects the line segment $BE$. [img]https://cdn.artofproblemsolving.com/attachments/3/2/5ce44f1e091786b865ae4319bda56c3ddbb8d7.png[/img]

2017 Regional Olympiad of Mexico West, 2

Tags: geometry , bisects segment

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 $SQ$.

2018 JBMO Shortlist, G1

Tags: bisects segment , geometry

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 $CH$.

2023 Iranian Geometry Olympiad, 4

Tags: geometry , bisects segment

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]

2021 Yasinsky Geometry Olympiad, 5

Tags: incenter , bisects segment , construction , ruler , geometry

Circle $\omega$ is inscribed in the $\vartriangle ABC$, with center $I$. Using only a ruler, divide segment $AI$ in half. (Grigory Filippovsky)

VI Soros Olympiad 1999 - 2000 (Russia), 8.4

Tags: geometry , bisects segment

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 $BC$.

2019 Tournament Of Towns, 4

Tags: angle bisector , geometry , bisects segment , perpendicular

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)

2018 Saudi Arabia BMO TST, 1

Tags: radical axis , geometry , bisects segment

Let $ABC$ be a triangle with $M, N, P$ as midpoints of the segments $BC, CA,AB$ respectively. Suppose that $I$ is the intersection of angle bisectors of $\angle BPM, \angle MNP$ and $J$ is the intersection of angle bisectors of $\angle CN M, \angle MPN$. Denote $(\omega_1)$ as the circle of center $I$ and tangent to $MP$ at $D$, $(\omega_2)$ as the circle of center $J$ and tangent to $MN$ at $E$. a) Prove that $DE$ is parallel to $BC$. b) Prove that the radical axis of two circles $(\omega_1), (\omega_2)$ bisects the segment $DE$.

2021 Saudi Arabia Training Tests, 16

Tags: bisects segment , geometry

Let $ABC$ be an acute, non-isosceles triangle with circumcenter $O$, incenter $I$ and $(I)$ tangent to $BC$, $CA$, $AB$ at $D, E, F$ respectively. Suppose that $EF$ cuts $(O)$ at $P, Q$. Prove that $(PQD)$ bisects segment $BC$.

2021 Saudi Arabia Training Tests, 3

Tags: geometry , bisects segment , perpendicular bisector

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 $AO$, $BC$.

Croatia MO (HMO) - geometry, 2013.3

Tags: geometry , parallelogram , bisects , cyclic quadrilateral , concyclic , bisects segment

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 $CG$.

2011 Dutch IMO TST, 5

Tags: bisects segment , angle bisector , geometry

Let $ABC$ be a triangle with $|AB|> |BC|$. Let $D$ be the midpoint of $AC$. Let $E$ be the intersection of the angular bisector of $\angle ABC$ and the line $AC$. Let $F$ be the point on $BE$ such that $CF$ is perpendicular to $BE$. Finally, let $G$ be the intersection of $CF$ and $BD$. Prove that $DF$ divides the line segment $EG$ into two equal parts.

2020 Novosibirsk Oral Olympiad in Geometry, 4

Tags: geometry , bisects segment

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.

Kyiv City MO Juniors 2003+ geometry, 2013.9.5

Tags: bisects segment , circles , geometry , tangent circle

The two circles ${{w} _ {1}}, \, \, {{w} _ {2}}$ touch externally at the point $Q$. The common external tangent of these circles is tangent to ${{w} _ {1}}$ at the point $B$, $BA$ is the diameter of this circle. A tangent to the circle ${{w} _ {2}} $ is drawn through the point $A$, which touches this circle at the point $C$, such that the points $B$ and $C$ lie in one half-plane relative to the line $AQ$. Prove that the circle ${{w} _ {1}}$ bisects the segment $C $. (Igor Nagel)

2019 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometry , incenter , arc midpoint , bisects segment , circumcircle

Let $ABC$ a triangle, $I$ the incenter, $D$ the contact point of the incircle with the side $BC$ and $E$ the foot of the bisector of the angle $A$. If $M$ is the midpoint of the arc $BC$ which contains the point $A$ of the circumcircle of the triangle $ABC$ and $\{F\} = DI \cap AM$, prove that $MI$ passes through the midpoint of $[EF]$.

2011 Dutch IMO TST, 5

Tags: bisects segment , angle bisector , geometry

Let $ABC$ be a triangle with $|AB|> |BC|$. Let $D$ be the midpoint of $AC$. Let $E$ be the intersection of the angular bisector of $\angle ABC$ and the line $AC$. Let $F$ be the point on $BE$ such that $CF$ is perpendicular to $BE$. Finally, let $G$ be the intersection of $CF$ and $BD$. Prove that $DF$ divides the line segment $EG$ into two equal parts.

Kyiv City MO Seniors 2003+ geometry, 2013.11.3

Tags: geometry , bisects segment , diameter , perpendicular

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)

2012 Tournament of Towns, 4

Tags: parallelogram , bisects segment , geometry , circles

A circle touches sides $AB, BC, CD$ of a parallelogram $ABCD$ at points $K, L, M$ respectively. Prove that the line $KL$ bisects the height of the parallelogram drawn from the vertex $C$ to $AB$.

2015 Thailand TSTST, 1

Tags: geometry , circumcircle , bisects segment , right angle

Let $O$ be the circumcenter of an acute $\vartriangle ABC$ which has altitude $AD$. Let $AO$ intersect the circumcircle of $\vartriangle BOC$ again at $X$. If $E$ and $F$ are points on lines $AB$ and $AC$ such that $\angle XEA = \angle XFA = 90^o$ , then prove that the line $DX$ bisects the segment $EF$.

1998 Switzerland Team Selection Test, 5

Tags: geometry , bisects segment , circles

Points $A$ and $B$ are chosen on a circle $k$. Let AP and $BQ$ be segments of the same length tangent to $k$, drawn on different sides of line $AB$. Prove that the line $AB$ bisects the segment $PQ$.

2009 Switzerland - Final Round, 5

Tags: geometry , bisects segment , incenter , incircle

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 $AD$ .

Kyiv City MO Seniors 2003+ geometry, 2013.10.4

Tags: bisects segment , circles , geometry , tangent circle

The two circles ${{w} _ {1}}, \, \, {{w} _ {2}}$ touch externally at the point $Q$. The common external tangent of these circles is tangent to ${{w} _ {1}}$ at the point $B$, $BA$ is the diameter of this circle. A tangent to the circle ${{w} _ {2}} $ is drawn through the point $A$, which touches this circle at the point $C$, such that the points $B$ and $C$ lie in one half-plane relative to the line $AQ$. Prove that the circle ${{w} _ {1}}$ bisects the segment $C $. (Igor Nagel)