Olympiad problems

2013 Junior Balkan Team Selection Tests - Romania, 4

Consider acute triangles $ABC$ and $BCD$, with $\angle BAC = \angle BDC$, such that $A$ and $D$ are on opposite sides of line $BC$. Denote by $E$ the foot of the perpendicular line to $AC$ through $B$ and by $F$ the foot of the perpendicular line to $BD$ through $C$. Let $H_1$ be the orthocenter of triangle $ABC$ and $H_2$ be the orthocenter of $BCD$. Show that lines $AD, EF$ and $H_1H_2$ are concurrent.

2001 Cuba MO, 2

Tags: geometry , square , perpendicular

Let $ABCD$ be a square. On the sides $BC$ and $CD$ the points $M$ and $K$ respectively, so that $MC = KD$. Let $P$ the intersection point of of segments $MD$ and $BK$. Prove that $AP \perp MK$.

2010 Saudi Arabia IMO TST, 2

Tags: perpendicular , geometry

Let $ABCD$ be a convex quadrilateral such that $\angle ABC = \angle ADC =135^o$ and $$AC^2 BD^2=2AB\cdot BC \cdot CD\cdot DA.$$ Prove that the diagonals of $ABCD$ are perpendicular.

Estonia Open Junior - geometry, 2020.1.5

Tags: geometry , perpendicular , regular , pentagon

A circle $c$ with center $A$ passes through the vertices $B$ and $E$ of a regular pentagon $ABCDE$. The line $BC$ intersects the circle $c$ for second time at point $F$. Prove that the lines $DE$ and $EF$ are perpendicular.

2016 Indonesia MO, 1

Tags: perpendicular , angle bisector , projection , geometry

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 $\angle FEH$ .

2015 Ukraine Team Selection Test, 6

Tags: geometry , concyclic , perpendicular , altitude

Given an acute triangle $ABC, H$ is the foot of the altitude drawn from the point $A$ on the line $BC, P$ and $K \ne H$ are arbitrary points on the segments $AH$ and$ BC$ respectively. Segments $AC$ and $BP$ intersect at point $B_1$, lines $AB$ and $CP$ at point $C_1$. Let $X$ and $Y$ be the projections of point $H$ on the lines $KB_1$ and $KC_1$, respectively. Prove that points $A, P, X$ and $Y$ lie on one circle.

2019 Oral Moscow Geometry Olympiad, 6

Tags: excircle , geometry , perpendicularity , perpendicular , excenter

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.

the 7th XMO, 1

Tags: geometry , perpendicular

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]

2020 Dutch IMO TST, 4

Tags: geometry , perpendicular

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 $AP \perp XY$.

2012 China Northern MO, 1

Tags: geometry , perpendicular , right triangle

As shown in figure, given right $\vartriangle ABC$ with $\angle C=90^o$. $I$ is the incenter. The line $BI$ intersects segment $AC$ at the point $D$ . The line passing through $D$ parallel to $AI$ intersects $BC$ at point $E$. The line $EI$ intersects segment $AB$ at point $F$. Prove that $DF \perp AI$. [img]https://cdn.artofproblemsolving.com/attachments/2/4/6fc94adb4ce12c3bf07948b8c57170ca01b256.png[/img]

2019 Canadian Mathematical Olympiad Qualification, 6

Tags: pentagon , concurrency , geometry , perpendicular

Pentagon $ABCDE$ is given in the plane. Let the perpendicular from $A$ to line $CD$ be $F$, the perpendicular from $B$ to $DE$ be $G$, from $C$ to $EA$ be $H$, from $D$ to $AB$ be $I$,and from $E$ to $BC$ be $J$. Given that lines $AF,BG,CH$, and $DI$ concur, show that they also concur with line $EJ$.

2006 Cuba MO, 9

Tags: geometry , perpendicular , equal angles

In the cyclic quadrilateral $ABCD$, the diagonals $AC$ and $BD$ intersect at $P$. Let $O$ be the center of the circumcircle $ABCD$, and $E$ a point of the extension of $OC$ beyond $C$. A parallel line to $CD$ is drawn through $E$ that cuts the extension of $OD$ beyonf $D$ at $F$. Let $Q$ be a point interior to $ABCD$, such that $\angle AFQ = \angle BEQ$ and $\angle FAQ = \angle EBQ$. Prove that $PQ \perp CD$.

2023 Czech-Polish-Slovak Junior Match, 1

Tags: perpendicular , geometry

Given a triangle $ABC$, $BC = 2 \cdot AC$. Point $M$ is the midpoint of side $ BC$ and point $D$ lies on $AB$, with $AD = 2 \cdot BD$. Prove that the lines $AM$ and $MD$ are perpendicular.

2025 Philippine MO, P7

Tags: geometry , perpendicular , arbitrary point , circumcircle , midpoint , parallel , diameter

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.

2018 Thailand TSTST, 3

Tags: geometry , fixed , perpendicular , fixed point

Let $BC$ be a chord not passing through the center of a circle $\omega$. Point $A$ varies on the major arc $BC$. Let $E$ and $F$ be the projection of $B$ onto $AC$, and of $C$ onto $AB$ respectively. The tangents to the circumcircle of $\vartriangle AEF$ at $E, F$ intersect at $P$. (a) Prove that $P$ is independent of the choice of $A$. (b) Let $H$ be the orthocenter of $\vartriangle ABC$, and let $T$ be the intersection of $EF$ and $BC$. Prove that $TH \perp AP$.

2015 Germany Team Selection Test, 2

Tags: geometry , circumcircle , incenter , perpendicular , orthogonal

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]

2001 Singapore Team Selection Test, 1

Tags: ratio , geometry , perpendicular

In the acute triangle $ABC$, let $D$ be the foot of the perpendicular from $A$ to $BC$, let $E$ be the foot of the perpendicular from $D$ to $AC$, and let $F$ be a point on the line segment $DE$. Prove that $AF$ is perpendicular to $BE$ if and only if $FE/FD = BD/DC$

2015 Indonesia MO, 6

Tags: geometry , equal segments , circumcircle , perpendicular

Let $ABC$ be an acute angled triangle with circumcircle $O$. Line $AO$ intersects the circumcircle of triangle $ABC$ again at point $D$. Let $P$ be a point on the side $BC$. Line passing through $P$ perpendicular to $AP$ intersects lines $DB$ and $DC$ at $E$ and $F$ respectively . Line passing through $D$ perpendicular to $BC$ intersects $EF$ at point $Q$. Prove that $EQ = FQ$ if and only if $BP = CP$.

Kyiv City MO Juniors 2003+ geometry, 2019.9.2

Tags: median , geometry , perpendicular

In a right triangle $ABC$, the lengths of the legs satisfy the condition: $BC =\sqrt2 AC$. Prove that the medians $AN$ and $CM$ are perpendicular. (Hilko Danilo)

Kyiv City MO Seniors Round2 2010+ geometry, 2011.10.4

Tags: geometry , perpendicular , parallel , tangent circle

Let two circles be externally tangent at point $C$, with parallel diameters $A_1A_2, B_1B_2$ (i.e. the quadrilateral $A_1B_1B_2A_2$ is a trapezoid with bases $A_1A_2$ and $B_1B_2$ or parallelogram). Circle with the center on the common internal tangent to these two circles, passes through the intersection point of lines $A_1B_2$ and $A_2B_1$ as well intersects those lines at points $M, N$. Prove that the line $MN$ is perpendicular to the parallel diameters $A_1A_2, B_1B_2$. (Yuri Biletsky)

1997 Abels Math Contest (Norwegian MO), 2a

Tags: geometry , sum , perpendicular , independent , equilateral

Let $P$ be an interior point of an equilateral triangle $ABC$, and let $Q,R,S$ be the feet of perpendiculars from $P$ to $AB,BC,CA$, respectively. Show that the sum $PQ+PR+PS$ is independent of the choice of $P$.

2016 Sharygin Geometry Olympiad, 3

Tags: geometry , locus , trapezoid , perpendicular

A trapezoid $ABCD$ and a line $\ell$ perpendicular to its bases $AD$ and $BC$ are given. A point $X$ moves along $\ell$. The perpendiculars from $A$ to $BX$ and from $D$ to $CX$ meet at point $Y$ . Find the locus of $Y$ . by D.Prokopenko

2006 Korea Junior Math Olympiad, 7

Tags: geometry , tangent , perpendicular , circles

A line through point $P$ outside of circle $O$ meets the said circle at $B,C$ ($PB < PC$). Let $PO$ meet circle $O$ at $Q,D$ (with $PQ < PD$). Let the line passing $Q$ and perpendicular to $BC$ meet circle $O$ at $A$. If $BD^2 = AD\cdot CP$, prove that $PA$ is a tangent to $O$.

2015 Saudi Arabia GMO TST, 3

Tags: orthocenter , circumcenter , perpendicular , geometry

Let $BD$ and $CE$ be altitudes of an arbitrary scalene triangle $ABC$ with orthocenter $H$ and circumcenter $O$. Let $M$ and $N$ be the midpoints of sides $AB$, respectively $AC$, and $P$ the intersection point of lines $MN$ and $DE$. Prove that lines $AP$ and $OH$ are perpendicular. Liana Topan

Kyiv City MO Juniors Round2 2010+ geometry, 2011.9.4

Tags: geometry , perpendicular , parallel , tangent circle

Let two circles be externally tangent at point $C$, with parallel diameters $A_1A_2, B_1B_2$ (i.e. the quadrilateral $A_1B_1B_2A_2$ is a trapezoid with bases $A_1A_2$ and $B_1B_2$ or parallelogram). Circle with the center on the common internal tangent to these two circles, passes through the intersection point of lines $A_1B_2$ and $A_2B_1$ as well intersects those lines at points $M, N$. Prove that the line $MN$ is perpendicular to the parallel diameters $A_1A_2, B_1B_2$. (Yuri Biletsky)