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.

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Found problems: 509

Denmark (Mohr) - geometry, 2015.3
Tags: geometry , equilateral , equal segments
Triangle $ABC$ is equilateral. The point $D$ lies on the extension of $AB$ beyond $B$, the point $E$ lies on the extension of $CB$ beyond $B$, and $|CD| = |DE|$. Prove that $|AD| = |BE|$. [img]https://1.bp.blogspot.com/-QnAXFw3ijn0/XzR0YjqBQ3I/AAAAAAAAMU0/0TvhMQtBNjolYHtgXsQo2OPGJzEYSfCwACLcBGAsYHQ/s0/2015%2BMohr%2Bp3.png[/img]

Kyiv City MO Juniors 2003+ geometry, 2016.8.51
Tags: equal segments , geometry , equal angles
In the quadrilateral $ABCD$, shown in fig. , the equations are true: $\angle ABC = \angle BCD$ and $2AB = CD$. On the side $BC$, a point $X$ is selected such that $\angle BAX = \angle CDA$. Prove that $AX = AD$. [img]https://cdn.artofproblemsolving.com/attachments/2/9/0884eb311d1e40300c1e5980fd53eaadfa7a25.png[/img]

2022 Austrian MO National Competition, 2
Tags: geometry , equal segments , orthocenter
Let $ ABC$ be an acute-angled, non-isosceles triangle with orthocenter $H$, $M$ midpoint of side $AB$ and $w$ bisector of angle $\angle ACB$. Let $S$ be the point of intersection of the perpendicular bisector of side $AB$ with $w$ and $F$ the foot of the perpendicular from $H$ on $w$. Prove that the segments $MS$ and $MF$ are equal. [i](Karl Czakler)[/i]

2004 Estonia Team Selection Test, 2
Tags: equal segments , circumcenter , parallel , geometry
Let $O$ be the circumcentre of the acute triangle $ABC$ and let lines $AO$ and $BC$ intersect at point $K$. On sides $AB$ and $AC$, points $L$ and $M$ are chosen such that $|KL|= |KB|$ and $|KM| = |KC|$. Prove that segments $LM$ and $BC$ are parallel.

2011 Junior Balkan Team Selection Tests - Moldova, 3
Tags: equal segments , geometry , angle , circles
Let $ABC$ be a triangle with $ \angle ACB = 90^o + \frac12 \angle ABC$ . The point $M$ is the midpoint of the side $BC$ . A circle with center at vertex $A$ intersects the line $BC$ at points $M$ and $D$. Prove that $MD = AB$.

2008 Mathcenter Contest, 2
Tags: equal segments , geometry , incircle
In triangle $ABC$ ($AB\not= AC$), the incircle is tangent to the sides of $BC$ ,$CA$ , $AB$ at $D$ ,$E$, $F$ respectively. Let $AD$ meet the incircle again at point $P$, let $EF$ and the line passing through the point $P$ and perpendicular to $AD$ intersect at $Q$. Let $AQ$ intersect $DE$ at $X$ and $DF$ at $Y$. Prove that $AX=AY$. [i](tatari/nightmare)[/i]

2015 Caucasus Mathematical Olympiad, 5
Tags: equal segments , geometry , altitude , midpoint , equal angles
Let $AA_1$ and $CC_1$ be the altitudes of the acute-angled triangle $ABC$. Let $K,L$ and $M$ be the midpoints of the sides $AB,BC$ and $CA$ respectively. Prove that if $\angle C_1MA_1 =\angle ABC$, then $C_1 K = A_1L$.

Ukraine Correspondence MO - geometry, 2008.7
Tags: equal segments , equal angles , geometry
On the sides $AC$ and $AB$ of the triangle $ABC$, the points $D$ and $E$ were chosen such that $\angle ABD =\angle CBD$ and $3 \angle ACE = 2\angle BCE$. Let $H$ be the point of intersection of $BD$ and $CE$, and $CD = DE = CH$. Find the angles of triangle $ABC$.

2017 Switzerland - Final Round, 5
Tags: equal segments , geometry , circumcircle , tangent
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 $BR = 2XY$.

2020 Ukrainian Geometry Olympiad - December, 5
Tags: equal segments , geometry , circumcenter
Let $O$ is the center of the circumcircle of the triangle $ABC$. We know that $AB =1$ and $AO = AC = 2$ . Points $D$ and $E$ lie on extensions of sides $AB$ and $AC$ beyond points $B$ and $C$ respectively such that $OD = OE$ and $BD =\sqrt2 EC$. Find $OD^2$.

2014 Gulf Math Olympiad, 3
Tags: equal segments , geometry , right triangle , semicircle , angle
(i) $ABC$ is a triangle with a right angle at $A$, and $P$ is a point on the hypotenuse $BC$. The line $AP$ produced beyond $P$ meets the line through $B$ which is perpendicular to $BC$ at $U$. Prove that $BU = BA$ if, and only if, $CP = CA$. (ii) $A$ is a point on the semicircle $CB$, and points $X$ and $Y$ are on the line segment $BC$. The line $AX$, produced beyond $X$, meets the line through $B$ which is perpendicular to $BC$ at $U$. Also the line $AY$, produced beyond $Y$, meets the line through $C$ which is perpendicular to $BC$ at $V$. Given that $BY = BA$ and $CX = CA$, determine the angle $\angle VAU$.

2015 Indonesia MO, 6
Tags: circumcircle , geometry , equal segments , 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 $BP = CP$.

1984 Tournament Of Towns, (054) O2
Tags: equal segments , convex , pentagon , geometry
In the convex pentagon $ABCDE$, $AE = AD$, $AB = AC$, and angle $CAD$ equals the sum of angles $AEB$ and $ABE$. Prove that segment $CD$ is double the length of median $AM$ of triangle $ABE$.

Indonesia MO Shortlist - geometry, g10
Tags: geometry , equal segments , isosceles
Given a triangle $ABC$ with $AB = AC$, angle $\angle A = 100^o$ and $BD$ bisector of angle $\angle B$. Prove that $$BC = BD + DA.$$

2014 Contests, 2
Tags: equal segments , parallelogram , geometry
Let $ABCD$ be a parallelogram. On side $AB$, point $M$ is taken so that $AD = DM$. On side $AD$ point $N$ is taken so that $AB = BN$. Prove that $CM = CN$.

2005 Bosnia and Herzegovina Junior BMO TST, 4
Tags: geometry , trapezoid , equal segments
The sum of the angles on the bigger base of a trapezoid is $90^o$. Prove that the line segment whose ends are the midpoints of the bases, is equal to the line segment whose ends are the midpoints of the diagonals.

2021 Saudi Arabia Training Tests, 10
Tags: geometry , equal segments , tangent circle
Let $AB$ be a chord of the circle $(O)$. Denote M as the midpoint of the minor arc $AB$. A circle $(O')$ tangent to segment $AB$ and internally tangent to $(O)$. A line passes through $M$, perpendicular to $O'A$, $O'B$ and cuts $AB$ respectively at $C, D$. Prove that $AB = 2CD$.

2014 Estonia Team Selection Test, 4
Tags: geometry , circumcircle , altitude , equal segments
In an acute triangle the feet of altitudes drawn from vertices $A$ and $B$ are $D$ and $E$, respectively. Let $M$ be the midpoint of side $AB$. Line $CM$ intersects the circumcircle of $CDE$ again in point $P$ and the circumcircle of $CAB$ again in point $Q$. Prove that $|MP| = |MQ|$.

1974 All Soviet Union Mathematical Olympiad, 198
Tags: equal segments , right isosceles
Given points $D$ and $E$ on the legs $[CA]$ and $[CB]$, respectively, of the isosceles right triangle. $|CD| = |CE|$. The extensions of the perpendiculars from $D$ and $C$ to the line $AE$ cross the hypotenuse $AB$ in the points $K$ and $L$. Prove that $|KL| = |LB|$

Ukrainian From Tasks to Tasks - geometry, 2015.14
Tags: concyclic , equal segments , geometry
On the side $AB$ of the triangle $ABC$ mark the points $M$ and $N$, such that $BM = BC$ and $AN = AC$. Then on the sides $BC$ and $AC$ mark the points$ P$ and $Q$, respectively, such that $BP = BN$ and $AQ = AM$. Prove that the points $C, Q, M, N$ and $P$ lie on the same circle.

1989 All Soviet Union Mathematical Olympiad, 501
Tags: rotation , equal segments , midpoint , collinear
$ABCD$ has $AB = CD$, but $AB$ not parallel to $CD$, and $AD$ parallel to $BC$. The triangle is $ABC$ is rotated about $C$ to $A'B'C$. Show that the midpoints of $BC, B'C$ and $A'D$ are collinear.

2012 Estonia Team Selection Test, 4
Tags: equal segments , geometry , circumcenter , circumcircle , isosceles
Let $ABC$ be a triangle where $|AB| = |AC|$. Points $P$ and $Q$ are different from the vertices of the triangle and lie on the sides $AB$ and $AC$, respectively. Prove that the circumcircle of the triangle $APQ$ passes through the circumcenter of $ABC$ if and only if $|AP| = |CQ|$.

2023 China Western Mathematical Olympiad, 3
Tags: equal segments , equal angles , geometry
In $\triangle ABC$, points $P,Q$ satisfy $\angle PBC = \angle QBA$ and $\angle PCB = \angle QCA$, $D$ is a point on $BC$ such that $\angle PDB=\angle QDC$. Let $X,Y$ be the reflections of $A$ with respect to lines $BP$ and $CP$, respectively. Prove that $DX=DY$. [img]https://cdn.artofproblemsolving.com/attachments/a/7/f208f1651afc0fef9eef4c68ba36bf77556058.jpg[/img]

2008 Federal Competition For Advanced Students, P1, 4
Tags: equal segments , isosceles , midpoint , isosceles triangle , geometry
In a triangle $ABC$ let $E$ be the midpoint of the side $AC$ and $F$ the midpoint of the side $BC$. Let $G$ be the foot of the perpendicular from $C$ to $ AB$. Show that $\vartriangle EFG$ is isosceles if and only if $\vartriangle ABC$ is isosceles.

Champions Tournament Seniors - geometry, 2018.3
Tags: equal angles , equal segments , geometry , parallelogram
The vertex $F$ of the parallelogram $ACEF$ lies on the side $BC$ of parallelogram $ABCD$. It is known that $AC = AD$ and $AE = 2CD$. Prove that $\angle CDE = \angle BEF$.