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: 351

2010 District Olympiad, 3

Consider triangle $ABC$ with $AB = AC$ and $\angle A = 40 ^o$. The points $S$ and $T$ are on the sides $AB$ and $BC$, respectively, so that $\angle BAT = \angle BCS= 10 ^o$. The lines $AT$ and $CS$ intersect at point $P$. Prove that $BT = 2PT$.

2000 Tournament Of Towns, 2

$ABCD$ is parallelogram, $M$ is the midpoint of side $CD$ and $H$ is the foot of the perpendicular from $B$ to line $AM$. Prove that $BCH$ is an isosceles triangle. (M Volchkevich)

1968 Dutch Mathematical Olympiad, 1

On the base $AB$ of the isosceles triangle $ABC$, lies the point $P$ such that $AP : PB = 1 : 2$. Determine the minimum of $\angle ACP$.

Estonia Open Junior - geometry, 1999.1.2

Two different points $X$ and $Y$ are chosen in the plane. Find all the points $Z$ in this plane for which the triangle $XYZ$ is isosceles.

2011 Oral Moscow Geometry Olympiad, 4

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

2014 Romania National Olympiad, 4

Outside the square $ABCD$ is constructed the right isosceles triangle $ABD$ with hypotenuse $[AB]$. Let $N$ be the midpoint of the side $[AD]$ and ${M} = CE \cap AB$, ${P} = CN \cap AB$ , ${F} = PE \cap MN$. On the line $FP$ the point $Q$ is considered such that the $[CE$ is the bisector of the angle $QCB$. Prove that $MQ \perp CF$.

2022 Novosibirsk Oral Olympiad in Geometry, 5

Tags: area , geometry , isosceles
Two isosceles triangles of the same area are located as shown in the figure. Find the angle $x$. [img]https://cdn.artofproblemsolving.com/attachments/a/6/f7dbfd267274781b67a5f3d5a9036fb2905156.png[/img]

2014 Contests, 1

As shown in the figure, given $\vartriangle ABC$ with $\angle B$, $\angle C$ acute angles, $AD \perp BC$, $DE \perp AC$, $M$ midpoint of $DE$, $AM \perp BE$. Prove that $\vartriangle ABC$ is isosceles. [img]https://cdn.artofproblemsolving.com/attachments/a/8/f553c33557979f6f7b799935c3bde743edcc3c.png[/img]

2000 Estonia National Olympiad, 4

Let $E$ be the midpoint of the side $AB$ of the parallelogram $ABCD$. Let $F$ be the projection of $B$ on $AC$. Prove that the triangle $ABF$ is isosceles

2021 New Zealand MO, 5

Let $ABC$ be an isosceles triangle with $AB = AC$. Point $D$ lies on side $AC$ such that $BD$ is the angle bisector of $\angle ABC$. Point $E$ lies on side $BC$ between $B$ and $C$ such that $BE = CD$. Prove that $DE$ is parallel to $AB$.

2014 Contests, 2 seniors

On the sides of triangle $ABC$, isosceles right-angled triangles $AUB, CVB$, and $AWC$ are placed. These three triangles have their right angles at vertices $U, V$ , and $W$, respectively. Triangle $AUB$ lies completely inside triangle $ABC$ and triangles $CVB$ and $AWC$ lie completely outside $ABC$. See the figure. Prove that quadrilateral $UVCW$ is a parallelogram. [asy] import markers; unitsize(1.5 cm); pair A, B, C, U, V, W; A = (0,0); B = (2,0); C = (1.7,2.5); U = (B + rotate(90,A)*(B))/2; V = (B + rotate(90,C)*(B))/2; W = (C + rotate(90,A)*(C))/2; draw(A--B--C--cycle); draw(A--W, StickIntervalMarker(1,1,size=2mm)); draw(C--W, StickIntervalMarker(1,1,size=2mm)); draw(B--V, StickIntervalMarker(1,2,size=2mm)); draw(C--V, StickIntervalMarker(1,2,size=2mm)); draw(A--U, StickIntervalMarker(1,3,size=2mm)); draw(B--U, StickIntervalMarker(1,3,size=2mm)); draw(rightanglemark(A,U,B,5)); draw(rightanglemark(B,V,C,5)); draw(rightanglemark(A,W,C,5)); dot("$A$", A, S); dot("$B$", B, S); dot("$C$", C, N); dot("$U$", U, NE); dot("$V$", V, NE); dot("$W$", W, NW); [/asy]

2012 Bundeswettbewerb Mathematik, 4

From the vertices of a regular 27-gon, seven are chosen arbitrarily. Prove that among these seven points there are three points that form an isosceles triangle or four points that form an isosceles trapezoid.

2011 Sharygin Geometry Olympiad, 3

Restore the isosceles triangle $ABC$ ($AB = AC$) if the common points $I, M, H$ of bisectors, medians and altitudes respectively are given.

2018 Yasinsky Geometry Olympiad, 4

Let $I_a$ be the point of the center of an ex-circle of the triangle $ABC$, which touches the side $BC$ . Let $W$ be the intersection point of the bisector of the angle $\angle A$ of the triangle $ABC$ with the circumcircle of the triangle $ABC$. Perpendicular from the point $W$ on the straight line $AB$, intersects the circumcircle of $ABC$ at the point $P$. Prove, that if the points $B, P, I_a$ lie on the same line, then the triangle $ABC$ is isosceles. (Mykola Moroz)

May Olympiad L2 - geometry, 2009.2

Let $ABCD$ be a convex quadrilateral such that the triangle $ABD$ is equilateral and the triangle $BCD$ is isosceles, with $\angle C = 90^o$. If $E$ is the midpoint of the side $AD$, determine the measure of the angle $\angle CED$.

Novosibirsk Oral Geo Oly VII, 2019.4

Two squares and an isosceles triangle are positioned as shown in the figure (the up left vertex of the large square lies on the side of the triangle). Prove that points $A, B$ and $C$ are collinear. [img]https://cdn.artofproblemsolving.com/attachments/d/c/03515e40f74ced1f8243c11b3e610ef92137ac.png[/img]

2006 Spain Mathematical Olympiad, 3

$ABC$ is an isosceles triangle with $AB = AC$. Let $P$ be any point of a circle tangent to the sides $AB$ in $B$ and to AC in C. Denote $a$, $b$ and $c$ to the distances from $P$ to the sides $BC, AC$ and $AB$ respectively. Prove that: $a^2=bc$

2017 Singapore Junior Math Olympiad, 3

Let $ABC$ be a triangle with $AB=AC$. Let $D$ be a point on $BC$, and $E$ a point on $AD$ such that $\angle BED=\angle BAC=2\angle CED$. Prove that $BD=2CD$.

2023 Dutch IMO TST, 3

The center $O$ of the circle $\omega$ passing through the vertex $C$ of the isosceles triangle $ABC$ ($AB = AC$) is the interior point of the triangle $ABC$. This circle intersects segments $BC$ and $AC$ at points $D \ne C$ and $E \ne C$, respectively, and the circumscribed circle $\Omega$ of the triangle $AEO$ at the point $F \ne E$. Prove that the center of the circumcircle of the triangle $BDF$ lies on the circle $\Omega$.

Kyiv City MO Juniors 2003+ geometry, 2018.8.3

In the isosceles triangle $ABC$ with the vertex at the point $B$, the altitudes $BH$ and $CL$ are drawn. The point $D$ is such that $BDCH$ is a rectangle. Find the value of the angle $DLH$. (Bogdan Rublev)

2005 Thailand Mathematical Olympiad, 3

Triangle $\vartriangle ABC$ is isosceles with $AB = AC$ and $\angle ABC = 2\angle BAC$. Compute $\frac{AB}{BC}$ .

2015 Singapore Junior Math Olympiad, 4

Let $A$ be a set of numbers chosen from $1,2,..., 2015$ with the property that any two distinct numbers, say $x$ and $y$, in $A$ determine a unique isosceles triangle (which is non equilateral) whose sides are of length $x$ or $y$. What is the largest possible size of $A$?

1992 Poland - Second Round, 3

Through the center of gravity of the acute-angled triangle $ ABC $, lines are drawn perpendicular to the sides $ BC $, $ CA $, $ AB $, intersecting them at the points $ P $, $ Q $, $ R $, respectively. Prove that if $ |BP|\cdot |CQ| \cdot |AR| = |PC| \cdot |QA| \cdot |RB| $, then the triangle $ ABC $ is isosceles. Note: According to Ceva's theorem, the assumed equality of products is equivalent to the fact that the lines $ AP $, $ BQ $, $ CR $ have a common point.

1968 German National Olympiad, 6

Prove the following two statements: (a) If a triangle is isosceles, then two of its bisectors are of equal length. (b) If two angle bisectors in a triangle are of equal length, then it is isosceles.

Durer Math Competition CD Finals - geometry, 2021.C3

In the isosceles triangle $ABC$ we have $AC = BC$. Let $X$ be an arbitrary point of the segment $AB$. The line parallel to $BC$ and passing through $X$ intersects the segment $AC$ in $N$, and the line parallel to $AC$ and passing through $BC$ intersects the segment $BC$ in $M$. Let $k_1$ be the circle with center $N$ and radius $NA$. Similarly, let $k_2$ be the circle with center $M$ and radius $MB$. Let $T$ be the intersection of the circles $k_1$ and $k_2$ different from $X$. Show that the angles $\angle NCM$ and $\angle NTM$ are equal.