This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 509

2012 Denmark MO - Mohr Contest, 5
Tags: hexagon , equal angles , equilateral , equal segments , geometry
In the hexagon $ABCDEF$, all angles are equally large. The side lengths satisfy $AB = CD = EF = 3$ and $BC = DE = F A = 2$. The diagonals $AD$ and $CF$ intersect each other in the point $G$. The point $H$ lies on the side $CD$ so that $DH = 1$. Prove that triangle $EGH$ is equilateral.

2020 Tournament Of Towns, 5
Tags: right angle , trapezoid , equal segments , trapezium , geometry
Let $ABCD$ be an inscribed trapezoid. The base $AB$ is $3$ times longer than $CD$. Tangents to the circumscribed circle at the points $A$ and $C$ intersect at the point $K$. Prove that the angle $KDA$ is a right angle. Alexandr Yuran

Estonia Open Senior - geometry, 1994.2.2
Tags: geometry , area , cyclic , cyclic quadrilateral , equal segments
The two sides $BC$ and $CD$ of an inscribed quadrangle $ABCD$ are of equal length. Prove that the area of this quadrangle is equal to $S =\frac12 \cdot AC^2 \cdot \sin \angle A$

2008 Abels Math Contest (Norwegian MO) Final, 4b
Tags: geometry , equal segments , circumcenter
A point $D$ lies on the side $BC$ , and a point $E$ on the side $AC$ , of the triangle $ABC$ , and $BD$ and $AE$ have the same length. The line through the centres of the circumscribed circles of the triangles $ADC$ and $BEC$ crosses $AC$ in $K$ and $BC$ in $L$. Show that $KC$ and $LC$ have the same length.

Kyiv City MO Seniors Round2 2010+ geometry, 2019.11.3.1
Tags: geometry , incenter , centroid , equal segments
It is known that in the triangle $ABC$ the smallest side is $BC$. Let $X, Y, K$ and $L$ - points on the sides $AB, AC$ and on the rays $CB, BC$, respectively, are such that $BX = BK = BC =CY =CL$. The line $KX$ intersects the line $LY$ at the point $M$. Prove that the intersection point of the medians $\vartriangle KLM$ coincides with the center of the inscribed circle $\vartriangle ABC$.

2015 Caucasus Mathematical Olympiad, 3
Tags: equal angles , equal segments , circumcircle , angle bisector , geometry
Let $AL$ be the angle bisector of the acute-angled triangle $ABC$. and $\omega$ be the circle circumscribed about it. Denote by $P$ the intersection point of the extension of the altitude $BH$ of the triangle $ABC$ with the circle $\omega$ . Prove that if $\angle BLA= \angle BAC$, then $BP = CP$.

Swiss NMO - geometry, 2017.1
Tags: geometry , angle bisector , circles , equal segments
Let $A$ and $B$ be points on the circle $k$ with center $O$, so that $AB> AO$. Let $C$ be the intersection of the bisectors of $\angle OAB$ and $k$, different from $A$. Let $D$ be the intersection of the straight line $AB$ with the circumcircle of the triangle $OBC$, different from $B$. Show that $AD = AO$ .

2021 Austrian Junior Regional Competition, 2
Tags: geometry , equal segments , circumcenter
A triangle $ABC$ with circumcenter $U$ is given, so that $\angle CBA = 60^o$ and $\angle CBU = 45^o$ apply. The straight lines $BU$ and $AC$ intersect at point $D$. Prove that $AD = DU$. (Karl Czakler)

1982 Swedish Mathematical Competition, 4
Tags: geometry , equal segments
$ABC$ is a triangle with $AB = 33$, $AC = 21$ and $BC = m$, an integer. There are points $D$, $E$ on the sides $AB$, $AC$ respectively such that $AD = DE = EC = n$, an integer. Find $m$.

2022 Federal Competition For Advanced Students, P1, 2
Tags: geometry , cyclic quadrilateral , equal segments
The points $A, B, C, D$ lie in this order on a circle with center $O$. Furthermore, the straight lines $AC$ and $BD$ should be perpendicular to each other. The base of the perpendicular from $O$ on $AB$ is $F$. Prove $CD = 2 OF$. [i](Karl Czakler)[/i]

2013 Oral Moscow Geometry Olympiad, 4
Tags: diagonal , geometry , similar triangles , perpendicular , equal segments
Similar triangles $ABM, CBP, CDL$ and $ADK$ are built on the sides of the quadrilateral $ABCD$ with perpendicular diagonals in the outer side (the neighboring ones are oriented differently). Prove that $PK = ML$.

2007 Thailand Mathematical Olympiad, 5
Tags: geometry , equal segments
A triangle $\vartriangle ABC$ has $\angle A = 90^o$, and a point $D$ is chosen on $AC$. Point $F$ is the foot of altitude from $A$ to $BC$. Suppose that $BD = DC = CF = 2$. Compute $AC$.

Kyiv City MO Juniors 2003+ geometry, 2011.9.41
Tags: tangent , geometry , circumcircle , equal segments
The triangle $ABC$ is inscribed in a circle. At points $A$ and $B$ are tangents to this circle, which intersect at point $T$. A line drawn through the point $T$ parallel to the side $AC$ intersects the side $BC$ at the point $D$. Prove that $AD = CD$.

2000 Tournament Of Towns, 2
Tags: equal segments , circumcenter , circumcircle , chord , geometry
The chords $AC$ and $BD$ of a, circle with centre $O$ intersect at the point $K$. The circumcentres of triangles $AKB$ and $CKD$ are $M$ and $N$ respectively. Prove that $OM = KN$. (A Zaslavsky )

2019 Saudi Arabia Pre-TST + Training Tests, 1.3
Tags: geometry , trapezoid , right angle , incircle , equal segments
Let $ABCD$ be a trapezoid with $\angle A = \angle B = 90^o$ and a point $E$ lies on the segment $CD$. Denote $(\omega)$ as incircle of triangle $ABE$ and it is tangent to $AB,AE,BE$ respectively at $P, F,K$. Suppose that $KF$ cuts $BC,AD$ at $M,N$ and $PM,PN$ cut $(\omega)$ at $H, T$. Prove that $PH = PT$.

2005 Paraguay Mathematical Olympiad, 5
Tags: chord , geometry , equal segments
Given a chord $PQ$ of a circle and $M$ the midpoint of the chord, let $AB$ and $CD$ be two chords that pass through $M$. $AC$ and $BD$ are drawn until $PQ$ is intersected at points $X$ and $Y$ respectively. Show that $X$ and $Y$ are equidistant from $M$.

2021 Novosibirsk Oral Olympiad in Geometry, 3
Tags: geometry , equal segments
Prove that in a triangle one of the sides is twice as large as the other if and only if a median and an angle bisector of this triangle are perpendicular

2016 Saudi Arabia IMO TST, 2
Tags: convex , hexagon , equal segments , equal angles , geometry , angle
Let $ABCDEF$ be a convex hexagon with $AB = CD = EF$, $BC =DE = FA$ and $\angle A+\angle B = \angle C +\angle D = \angle E +\angle F$. Prove that $\angle A=\angle C=\angle E$ and $\angle B=\angle D=\angle F$. Tran Quang Hung

Swiss NMO - geometry, 2017.5
Tags: circumcircle , equal segments , tangent , geometry
Let $ABC$ be a triangle with $AC> AB$. Let $P$ be the intersection of $BC$ and the tangent through $A$ around the triangle $ABC$. Let $Q$ be the point on the straight line $AC$, so that $AQ = AB$ and $A$ is between $C$ and $Q$. Let $X$ and $Y$ be the center of $BQ$ and $AP$. Let $R$ be the point on $AP$ so that $AR = BP$ and $R$ is between $A$ and $P$. Show that $BR = 2XY$.

OMMC POTM, 2024 11
Tags: geometry , equal segments
Rectangle $ABCD$ with $AB>BC$ has point $P$ inside of it and $Q$ outside of it, such that $PQCD$ is a parallelogram with $PD=AD$. Let $M$ be the midpoint of $CD$. Give that $\angle AMP=\angle BMQ$, prove that $AB=2BC$.

2013 Regional Competition For Advanced Students, 4
Tags: equal segments , equal angles , pentagon , geometry , symmetry
We call a pentagon [i]distinguished [/i] if either all side lengths or all angles are equal. We call it [i]very distinguished[/i] if in addition two of the other parts are equal. i.e. $5$ sides and $2$ angles or $2$ sides and $5$ angles.Show that every very distinguished pentagon has an axis of symmetry.

Kyiv City MO Juniors 2003+ geometry, 2014.7.4
Tags: geometry , angle bisector , equal segments
In the quadrilateral $ABCD$ the condition $AD = AB + CD$ is fulfilled. The bisectors of the angles $BAD$ and $ADC$ intersect at the point $P $, as shown in Fig. Prove that $BP = CP$. [img]https://cdn.artofproblemsolving.com/attachments/3/1/67268635aaef9c6dc3363b00453b327cbc01f3.png[/img] (Maria Rozhkova)

2000 239 Open Mathematical Olympiad, 3
Tags: geometry , incenter , altitude , equal segments
Let $ AA_1 $ and $ CC_1 $ be the altitudes of the acute-angled triangle $ ABC $. A line passing through the centers of the inscribed circles the triangles $ AA_1C $ and $ CC_1A $ intersect the sides of $ AB $ and $ BC $ triangle $ ABC $ at points $ X $ and $ Y $. Prove that $ BX = BY $.

Kyiv City MO Seniors Round2 2010+ geometry, 2018.10.3
Tags: geometry , circumcircle , equal segments
In the acute triangle $ABC$ the orthocenter $H$ and the center of the circumscribed circle $O$ were noted. The line $AO$ intersects the side $BC$ at the point $D$. A perpendicular drawn to the side $BC$ at the point $D$ intersects the heights from the vertices $B$ and $C$ of the triangle $ABC$ at the points $X$ and $Y$ respectively. Prove that the center of the circumscribed circle $\Delta HXY$ is equidistant from the points $B$ and $C$. (Danilo Hilko)

Kyiv City MO Juniors Round2 2010+ geometry, 2015.7.41
Tags: equal segments , geometry
The equal segments $AB$ and $CD$ intersect at the point $O$ and divide it by the relation $AO: OB = CO: OD = 1: 2 $. The lines $AD$ and $BC$ intersect at the point $M$. Prove that $DM = MB$.