Olympiad problems

2013 Korea Junior Math Olympiad, 5

Tags: geometry , circumcircle , acute , circumcenter , perpendicular , midpoint

In an acute triangle $\triangle ABC, \angle A > \angle B$. Let the midpoint of $AB$ be $D$, and let the foot of the perpendicular from $A$ to $BC$ be $E$, and $B$ from $CA$ be $F$. Let the circumcenter of $\triangle DEF$ be $O$. A point $J$ on segment $BE$ satisfies $\angle ODC = \angle EAJ$. Prove that $AJ \cap DC$ lies on the circumcircle of $\triangle BDE$.

2021 OMpD, 2

Tags: geometry , perpendicular , arc midpoint

Let $ABC$ be a triangle, $\Gamma$ its circumcircle and $D$ the midpoint of the arc $AC$ of $\Gamma$ that does not contain $B$. If $O$ is the center of $\Gamma$ and I is the incenter of $ABC$, prove that $OI$ is perpendicular to $BD$ if and only if $AB + BC = 2AC$.

2006 MOP Homework, 2

Tags: equal angles , orthocenter , perpendicular , right angle , projection , geometry

Points $P$ and $Q$ lies inside triangle $ABC$ such that $\angle ACP =\angle BCQ$ and $\angle CAP = \angle BAQ$. Denote by $D,E$, and $F$ the feet of perpendiculars from $P$ to lines $BC,CA$, and $AB$, respectively. Prove that if $\angle DEF = 90^o$, then $Q$ is the orthocenter of triangle $BDF$.

Croatia MO (HMO) - geometry, 2022.7

Tags: geometry , perpendicular

In the triangle $ABC$ holds $|AB| = |AC|$ and the inscribed circle touches the sides $\overline{BC}$, $\overline{AC}$ and $\overline{AB}$ at the points $D$, $E$ and $F$ respectively . The perpendicular from the point $D$ to the line $EF$ intersects the side $\overline{AB}$ at the point $G$, and the circles circumscribed around the triangles $AEF$ and $ABC$ intersect at the points $A $and $T$. Prove that the lines $T G$ and $T F$ are perpendicular.

Kyiv City MO Seniors Round2 2010+ geometry, 2011.10.4

Tags: perpendicular , parallel , tangent circle , geometry

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)

2016 Czech-Polish-Slovak Junior Match, 1

Tags: geometry , area , max , perpendicular

Let $AB$ be a given segment and $M$ be its midpoint. We consider the set of right-angled triangles $ABC$ with hypotenuses $AB$. Denote by $D$ the foot of the altitude from $C$. Let $K$ and $L$ be feet of perpendiculars from $D$ to the legs $BC$ and $AC$, respectively. Determine the largest possible area of the quadrilateral $MKCL$. Czech Republic

2004 BAMO, 2

Tags: construction , straightedge , geometry , perpendicular

A given line passes through the center $O$ of a circle. The line intersects the circle at points $A$ and $B$. Point $P$ lies in the exterior of the circle and does not lie on the line $AB$. Using only an unmarked straightedge, construct a line through $P$, perpendicular to the line $AB$. Give complete instructions for the construction and prove that it works.

Ukraine Correspondence MO - geometry, 2020.8

Tags: geometry , perpendicular

Let $ABC$ be an acute triangle, $D$ be the midpoint of $BC$. Bisectors of angles $ADB$ and $ADC$ intersect the circles circumscribed around the triangles $ADB$ and $ADC$ at points $E$ and $F$, respectively. Prove that $EF\perp AD$.

2011 Oral Moscow Geometry Olympiad, 4

Tags: geometry , trapezoid , isosceles , midpoint , altitude , perpendicular

In the trapezoid $ABCD, AB = BC = CD, CH$ is the altitude. Prove that the perpendicular from $H$ on $AC$ passes through the midpoint of $BD$.

2015 May Olympiad, 3

Tags: geometry , perpendicular , regular polygon

Let $ABCDEFGHI$ be a regular polygon of $9$ sides. The segments $AE$ and $DF$ intersect at $P$. Prove that $PG$ and $AF$ are perpendicular.

2018 Danube Mathematical Competition, 2

Tags: geometry , angle chasing , perpendicular , perpendicularity

Let $ABC$ be a triangle such that in its interior there exists a point $D$ with $\angle DAC = \angle DCA = 30^o$ and $ \angle DBA = 60^o$. Denote $E$ the midpoint of the segment $BC$, and take $F$ on the segment $AC$ so that $AF = 2FC$. Prove that $DE \perp EF$.

2018 Yasinsky Geometry Olympiad, 6

Tags: geometry , angle chasing , perpendicular , midpoint

In the quadrilateral $ABCD$, the points $E, F$, and $K$ are midpoints of the $AB, BC, AD$ respectively. Known that $KE \perp AB, K F \perp BC$, and the angle $\angle ABC = 118^o$. Find $ \angle ACD$ (in degrees).

2005 Oral Moscow Geometry Olympiad, 4

Tags: geometry , perpendicular , hexagon , equal segments , right angle

Given a hexagon $ABCDEF$, in which $AB = BC, CD = DE, EF = FA$, and angles $A$ and $C$ are right. Prove that lines $FD$ and $BE$ are perpendicular. (B. Kukushkin)

2001 Singapore MO Open, 1

Tags: geometry , parallelogram , orthocenter , perpendicular

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.

Novosibirsk Oral Geo Oly IX, 2020.3

Tags: geometry , perpendicular , equal segments

Point $P$ is chosen inside triangle $ABC$ so that $\angle APC+\angle ABC=180^o$ and $BC=AP.$ On the side $AB$, a point $K$ is chosen such that $AK = KB + PC$. Prove that $CK \perp AB$.

the 7th XMO, 1

Tags: perpendicular , geometry

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]

2010 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometry , incenter , angle bisector , incircle , perpendicular

Let $ABC$ be a triangle inscribed in the circle $(O)$. Let $I$ be the center of the circle inscribed in the triangle and $D$ the point of contact of the circle inscribed with the side $BC$. Let $M$ be the second intersection point of the bisector $AI$ with the circle $(O)$ and let $P$ be the point where the line $DM$ intersects the circle $(O)$ . Show that $PA \perp PI$.

2017 Yasinsky Geometry Olympiad, 2

Tags: 3-dimensional geometry , 3d geometry , tetrahedron , geometry , equal angles , perpendicular

In the tetrahedron $DABC, AB=BC, \angle DBC =\angle DBA$. Prove that $AC \perp DB$.

Swiss NMO - geometry, 2007.4

Tags: geometry , perpendicular , orthocenter

Let $ABC$ be an acute-angled triangle with $AB> AC$ and orthocenter $H$. Let $D$ the projection of $A$ on $BC$. Let $E$ be the reflection of $C$ wrt $D$. The lines $AE$ and $BH$ intersect at point $S$. Let $N$ be the midpoint of $AE$ and let $M$ be the midpoint of $BH$. Prove that $MN$ is perpendicular to $DS$.

2021 Azerbaijan EGMO TST, 4

Tags: geometry , circles , orthocenter , perpendicular , incenter

Let $ABC$ be an acute, non isosceles with $I$ is its incenter. Denote $D, E$ as tangent points of $(I)$ on $AB,AC$, respectively. The median segments respect to vertex $A$ of triangles $ABE$ and $ACD$ meet$ (I)$ at$ P,Q,$ respectively. Take points $M, N$ on the line $DE$ such that $AM \parallel BE$ and $AN \parallel C D$ respectively. a) Prove that $A$ lies on the radical axis of $(MIP)$ and $(NIQ)$. b) Suppose that the orthocenter $H$ of triangle $ABC$ lies on $(I)$. Prove that there exists a line which is tangent to three circles of center $A, B, C$ and all pass through $H$.

2013 Saudi Arabia Pre-TST, 4.4

Tags: geometry , perpendicular , isosceles

$\vartriangle ABC$ is a triangle, $M$ the midpoint of $BC, D$ the projection of $M$ on $AC$ and $E$ the midppoint of $MD$. Prove that the lines $AE,BD$ are orthogonal if and only if $AB = AC$.

2006 Tournament of Towns, 2

Tags: geometry , incenter , perpendicular

The incircle of the quadrilateral $ABCD$ touches $AB,BC, CD$ and $DA$ at $E, F,G$ and $H$ respectively. Prove that the line joining the incentres of triangles $HAE$ and $FCG$ is perpendicular to the line joining the incentres of triangles $EBF$ and $GDH$. (4)

2010 Bundeswettbewerb Mathematik, 3

Tags: concyclic , collinear , altitude , perpendicular , geometry , projection

Given an acute-angled triangle $ABC$. Let $CB$ be the altitude and $E$ a random point on the line $CD$. Finally, let $P, Q, R$ and $S$ are the projections of $D$ on the straight lines $AC, AE, BE$ and $BC$. Prove that the points $P, Q, R$ and $S$ lie either on a circle or on one straight line.

1957 Moscow Mathematical Olympiad, 358

Tags: combinatorial geometry , broken line , 3d geometry , perpendicular , geometry

The segments of a closed broken line in space are of equal length, and each three consecutive segments are mutually perpendicular. Prove that the number of segments is divisible by $6$.

2015 Indonesia MO Shortlist, G6

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 $BP = CP$.