Found problems: 412
2016 Saudi Arabia GMO TST, 3
Let $ABC$ be a triangle with incenter $I$ . Let $CI, BI$ intersect $AB, AC$ at $D, E$ respectively. Denote by $\Delta_b,\Delta_c$ the lines symmetric to the lines $AB, AC$ with respect to $CD, BE$ correspondingly. Suppose that $\Delta_b,\Delta_c$ meet at $K$.
a) Prove that $IK \perp BC$.
b) If $I \in (K DE)$, prove that $BD + C E = BC$.
2019 Kosovo Team Selection Test, 4
Given a rectangle $ABCD$ such that $AB = b > 2a = BC$, let $E$ be the midpoint of $AD$. On a line parallel to $AB$ through point $E$, a point $G$ is chosen such that the area of $GCE$ is
$$(GCE)= \frac12 \left(\frac{a^3}{b}+ab\right)$$
Point $H$ is the foot of the perpendicular from $E$ to $GD$ and a point $I$ is taken on the diagonal $AC$ such that the triangles $ACE$ and $AEI$ are similar. The lines $BH$ and $IE$ intersect at $K$ and the lines $CA$ and $EH$ intersect at $J$. Prove that $KJ \perp AB$.
2020 Dutch IMO TST, 4
Let $ABC$ be an acute-angled triangle and let $P$ be the intersection of the tangents at $B$ and $C$ of the circumscribed circle of $\vartriangle ABC$. The line through $A$ perpendicular on $AB$ and cuts the line perpendicular on $AC$ through $C$ at $X$. The line through $A$ perpendicular on $AC$ cuts the line perpendicular on $AB$ through $B$ at $Y$. Show that $AP \perp XY$.
Kyiv City MO Juniors 2003+ geometry, 2014.85
Given an equilateral $\Delta ABC$, in which ${{A} _ {1}}, {{B} _ {1}}, {{C} _ {1}}$ are the midpoint of the sides $ BC, \, \, AC, \, \, AB$ respectively. The line $l$ passes through the vertex $A$, we denote by $P, Q$ the projection of the points $B, C$ on the line $l$, respectively (the line $ l $ and the point $Q, \, \, A, \, \, P$ are located as shown in fig.). Denote by $T $ the intersection point of the lines ${{B} _ {1}} P$ and ${{C} _ {1}} Q$. Prove that the line ${{A} _ {1}} T$ is perpendicular to the line $l$.
[img]https://cdn.artofproblemsolving.com/attachments/4/b/61f2f4ec9e6b290dfcd47e9351110bebd3bd43.png[/img]
(Serdyuk Nazar)
2016 Indonesia MO, 1
Let $ABCD$ be a cyclic quadrilateral wih both diagonals perpendicular to each other and intersecting at point $O$. Let $E,F,G,H$ be the orthogonal projections of $O$ on sides $AB,BC,CD,DA$ respectively.
a. Prove that $\angle EFG + \angle GHE = 180^o$
b. Prove that $OE$ bisects angle $\angle FEH$ .
2018 Iranian Geometry Olympiad, 3
Find all possible values of integer $n > 3$ such that there is a convex $n$-gon in which, each diagonal is the perpendicular bisector of at least one other diagonal.
Proposed by Mahdi Etesamifard
Ukrainian TYM Qualifying - geometry, 2020.11
In the acute-angled triangle $ABC$, the segment $AP$ was drawn and the center was marked $O$ of the circumscribed circle. The circumcircle of triangle $ABP$ intersects the line $AC$ for the second time at point $X$, the circumcircle of the triangle $ACP$ intersects the line $AB$ for the second time at the point $Y$. Prove that the lines $XY$ and $PO$ are perpendicular if and only if $P$ is the foor of the bisector of the triangle $ABC$.
1994 Italy TST, 1
Given a circle $\gamma$ and a point $P$ inside it, find the maximum and minimum value of the sum of the lengths of two perpendicular chords of $\gamma$ passing through $P$.
2005 Cuba MO, 5
On the circumcircle of triangle $ABC$, point $P$ is taken in such a way that the perpendicular drawn by the point $P$ to the line $AC$ cuts the circle also at the point $Q$, the perpendicular drawn by the point $Q$ to the line $AB$ cuts the circle also at point R and the perpendicular drawn by point $R$ to the line BC cuts the circle also at the point $P$. Let $O$ be the center of this circle. Prove that $\angle POC = 90^o$ .
1982 IMO Longlists, 54
The right triangles $ABC$ and $AB_1C_1$ are similar and have opposite orientation. The right angles are at $C$ and $C_1$, and we also have $ \angle CAB = \angle C_1AB_1$. Let $M$ be the point of intersection of the lines $BC_1$ and $B_1C$. Prove that if the lines $AM$ and $CC_1$ exist, they are perpendicular.
Cono Sur Shortlist - geometry, 2009.G4
Let $AA _1$ and $CC_1$ be altitudes of an acute triangle $ABC$. Let $I$ and $J$ be the incenters of the triangles $AA_1C$ and $AC_1C$ respectively. The $C_1J$ and $A_1 I$ lines cut into $T$. Prove that lines $AT$ and $TC$ are perpendicular.
2010 Oral Moscow Geometry Olympiad, 2
Quadrangle $ABCD$ is inscribed in a circle. The perpendicular from the vertex $C$ on the bisector of $\angle ABD$ intersects the line $AB$ at the point $C_1$. The perpendicular from the vertex $B$ on the bisector of $\angle ACD$ intersects the line $CD$ at the point $B_1$. Prove that $B_1C_1 \parallel AD$.
2025 Philippine MO, P7
In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$, let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$, and let $F$ be the point on line $AC$ such that $FA = FE$. Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$.
(a) Show that $AP$ and $BR$ are perpendicular. \\
(b) Show that $FM$ and $BM$ are perpendicular.
2022 Yasinsky Geometry Olympiad, 3
Given a triangle $ABC$, in which the medians $BE$ and $CF$ are perpendicular. Let $M$ is the intersection point of the medians of this triangle, and $L$ is its Lemoine point (the intersection point of lines symmetrical to the medians with respect to the bisectors of the corresponding angles). Prove that $ML \perp BC$.
(Mykhailo Sydorenko)
1999 All-Russian Olympiad Regional Round, 9.8
In triangle $ABC$ ($AB > BC$), $K$ and $M$ are the midpoints of sides $AB$ and $AC$, $O$ is the point of intersection of the angle bisectors. Let $P$ be the intersection point of lines $KM$ and $CO$, and the point $Q$ is such that $QP \perp KM$ and $QM \parallel BO$. Prove that $QO \perp AC$.
1964 Poland - Second Round, 2
The circle is divided into four non-overlapping gaps $ AB $, $ BC $, $ CD $ and $ DA $. Prove that the segment joining the midpoints of the arcs $AB$ and $CD$ is perpendicular to the segment joining the midpoints of the arcs $BC$ and $DA$.
2015 Germany Team Selection Test, 2
Let $ABC$ be an acute triangle with the circumcircle $k$ and incenter $I$. The perpendicular through $I$ in $CI$ intersects segment $[BC]$ in $U$ and $k$ in $V$. In particular $V$ and $A$ are on different sides of $BC$. The parallel line through $U$ to $AI$ intersects $AV$ in $X$.
Prove: If $XI$ and $AI$ are perpendicular to each other, then $XI$ intersects segment $[AC]$ in its midpoint $M$.
[i](Notation: $[\cdot]$ denotes the line segment.)[/i]
2013 Junior Balkan Team Selection Tests - Romania, 3
Let $D$ be the midpoint of the side $[BC]$ of the triangle $ABC$ with $AB \ne AC$ and $E$ the foot of the altitude from $BC$. If $P$ is the intersection point of the perpendicular bisector of the segment line $[DE]$ with the perpendicular from $D$ onto the the angle bisector of $BAC$, prove that $P$ is on the Euler circle of triangle $ABC$.
2022 OMpD, 4
Let $ABCD$ be a cyclic quadrilateral and $M,N$ be the midpoints of $AB$, $CD$ respectively. The diagonals $AC$ and $BD$ intersect at $L$. Suppose that the circumcircle of $LMN$, with center $T$, intersects the circumcircle of $ABCD$ at two distinct points $X,Y$. If the line $MN$ intersects the line $XY$ at $S$ and the line $XM$ intersects the line $YN$ at $P$, prove that $PL$ is perpendicular to $ST$.
2011 Tournament of Towns, 4
Four perpendiculars are drawn from four vertices of a convex pentagon to the opposite sides. If these four lines pass through the same point, prove that the perpendicular from the fifth vertex to the opposite side also passes through this point.
XMO (China) 2-15 - geometry, 7.1
As shown in the figure, it is known that $BC = AC$ in $ABC$, $M$ is the midpoint of $AB$, points $D$ and $E$ lie on $AB$ satisfying $\angle DCE = \angle MCB$, the circumscribed circle of $\vartriangle BDC$ and the circumscribed circle of $\vartriangle AEC$ intersect at point $F$ (different from point $C$), point $H$ lies on $AB$ such that the straight line $CM$ bisects the line segment $HF$. Let the circumcenters of $\vartriangle HFE$ and $\vartriangle BFM$ be $O_1$ and $O_2$ respectively. Prove that $O_1O_2\perp CF$.
[img]https://cdn.artofproblemsolving.com/attachments/e/4/e8fc62735b8cfbd382e490617f26d335c46823.png[/img]
1998 Slovenia Team Selection Test, 2
A semicircle with center $O$ and diameter $AB$ is given. Point $M$ on the extension of $AB$ is taken so that $AM > BM$. A line through $M$ intersects the semicircle at $C$ and $D$ so that $CM < DM$. The circumcircles of triangles $AOD$ and $OBC$ meet again at point $K$. Prove that $OK$ and $KM$ are perpendicular
2014 Czech-Polish-Slovak Junior Match, 4
Point $M$ is the midpoint of the side $AB$ of an acute triangle $ABC$. Circle with center $M$ passing through point $ C$, intersects lines $AC ,BC$ for the second time at points $P,Q$ respectively. Point $R$ lies on segment $AB$ such that the triangles $APR$ and $BQR$ have equal areas. Prove that lines $PQ$ and $CR$ are perpendicular.
2015 Germany Team Selection Test, 2
Let $ABC$ be an acute triangle with the circumcircle $k$ and incenter $I$. The perpendicular through $I$ in $CI$ intersects segment $[BC]$ in $U$ and $k$ in $V$. In particular $V$ and $A$ are on different sides of $BC$. The parallel line through $U$ to $AI$ intersects $AV$ in $X$.
Prove: If $XI$ and $AI$ are perpendicular to each other, then $XI$ intersects segment $[AC]$ in its midpoint $M$.
[i](Notation: $[\cdot]$ denotes the line segment.)[/i]
2019 Oral Moscow Geometry Olympiad, 6
In the acute triangle $ABC$, the point $I_c$ is the center of excircle on the side $AB$, $A_1$ and $B_1$ are the tangency points of the other two excircles with sides $BC$ and $CA$, respectively, $C'$ is the point on the circumcircle diametrically opposite to point $C$. Prove that the lines $I_cC'$ and $A_1B_1$ are perpendicular.