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

May Olympiad L1 - geometry, 2022.5
Tags: geometry , isosceles
Vero had an isosceles triangle made of paper. Using scissors, he divided it into three smaller triangles and painted them blue, red and green. Having done so, he observed that: $\bullet$ with the blue triangle and the red triangle an isosceles triangle can be formed, $\bullet$ with the blue triangle and the green triangle an isosceles triangle can be formed, $\bullet$ with the red triangle and the green triangle an isosceles triangle can be formed. Show what Vero's triangle looked like and how he might have made the cuts to make this situation be possible.

Kyiv City MO Seniors 2003+ geometry, 2006.10.4
Tags: geometry , isosceles , parallel , incircle
A circle $\omega$ is inscribed in the acute-angled triangle $\vartriangle ABC$, which touches the side $BC$ at the point $K$. On the lines $AB$ and $AC$, the points $P$ and $Q$, respectively, are chosen so that $PK \perp AC$ and $QK \perp AB$. Denote by $M$ and $N$ the points of intersection of $KP$ and $KQ$ with the circle $\omega$. Prove that if $MN \parallel PQ$, then $\vartriangle ABC$ is isosceles. (S. Slobodyanyuk)

Novosibirsk Oral Geo Oly IX, 2023.3
Tags: geometry , equal segments , isosceles , angle
An isosceles triangle $ABC$ with base $AC$ is given. On the rays $CA$, $AB$ and $BC$, the points $D, E$ and $F$ were marked, respectively, in such a way that $AD = AC$, $BE = BA$ and $CF = CB$. Find the sum of the angles $\angle ADB$, $\angle BEC$ and $\angle CFA$.

2022 May Olympiad, 5
Tags: geometry , isosceles
Vero had an isosceles triangle made of paper. Using scissors, he divided it into three smaller triangles and painted them blue, red and green. Having done so, he observed that: $\bullet$ with the blue triangle and the red triangle an isosceles triangle can be formed, $\bullet$ with the blue triangle and the green triangle an isosceles triangle can be formed, $\bullet$ with the red triangle and the green triangle an isosceles triangle can be formed. Show what Vero's triangle looked like and how he might have made the cuts to make this situation be possible.

2006 Junior Tuymaada Olympiad, 1
Tags: geometry , right triangle , isosceles , equal segments
On the equal $ AC $ and $ BC $ of an isosceles right triangle $ ABC $ , points $ D $ and $ E $ are marked respectively, so that $ CD = CE $. Perpendiculars on the straight line $ AE $, passing through the points $ C $ and $ D $, intersect the side $ AB $ at the points $ P $ and $ Q $.Prove that $ BP = PQ $.

2017 Ecuador Juniors, 3
Tags: geometry , perpendicular , isosceles
Given an isosceles triangle $ABC$ with $AB = AC$. Let $O$ be the circumcenter of $ABC$, $D$ the midpoint of $AB$ and $E$ the centroid of $ACD$. Prove that $CD \perp EO$.

2021 New Zealand MO, 5
Tags: parallelogram , isosceles , geometry
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 $AB$.

Denmark (Mohr) - geometry, 1994.5
Tags: equal area , right triangle , isosceles , area , geometry
In a right-angled and isosceles triangle, the two catheti are both length $1$. Find the length of the shortest line segment dividing the triangle into two figures with the same area, and specify the location of this line segment

2002 May Olympiad, 3
Tags: right triangle , isosceles , square , geometry
In a triangle $ABC$, right in $A$ and isosceles, let $D$ be a point on the side $AC$ ($A \ne D \ne C$) and $E$ be the point on the extension of $BA$ such that the triangle $ADE$ is isosceles. Let $P$ be the midpoint of segment $BD$, $R$ be the midpoint of the segment $CE$ and $Q$ the intersection point of $ED$ and $BC$. Prove that the quadrilateral $ARQP$ is a square.

2000 Tournament Of Towns, 2
Tags: isosceles , inradius , geometry
In triangle $ABC, AB = AC$. A line is drawn through $A$ parallel to $BC$. Outside triangle $ABC$, a circle is drawn tangent to this line, to the line $BC$, to $AB$ and to the incircle of $ABC$. If the radius of this circle is $1$ , determine the inradius of $ABC$. (RK Gordin)

Kyiv City MO Juniors Round2 2010+ geometry, 2017.7.41
Tags: geometry , isosceles , equal segments
Let $AC$ be the largest side of the triangle $ABC$. The point M is selected on the ray $AC$ ray, and point $N$ on ray $CA$ such that $CN = CB$ and$ AM = AB$ . a) Prove that $\vartriangle ABC$ is isosceles if we know that $BM = BN$. b) Will the statement remain true if $AC$ is not necessarily the largest side of triangle $ABC$?

2018 Yasinsky Geometry Olympiad, 4
Tags: geometry , angle chasing , isosceles , excircle , circumcircle
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)

2022 Austrian MO National Competition, 5
Tags: collinear , geometry , isosceles
Let $ABC$ be an isosceles triangle with base $AB$. We choose a point $P$ inside the triangle on altitude through $C$. The circle with diameter $CP$ intersects the straight line through $B$ and $P$ again at the point $D_P$ and the Straight through $A$ and $C$ one more time at point $E_P$. Prove that there is a point $F$ such that for any choice of $P$ the points $D_P , E_P$ and $F$ lie on a straight line. [i](Walther Janous)[/i]

2017 NZMOC Camp Selection Problems, 2
Tags: geometry , isosceles , parallelogram , equal segments
Let $ABCD$ be a parallelogram with an acute angle at $A$. Let $G$ be the point on the line $AB$, distinct from $B$, such that $CG = CB$. Let $H$ be the point on the line $BC$, distinct from $B$, such that $AB = AH$. Prove that triangle $DGH$ is isosceles.

2010 Junior Balkan Team Selection Tests - Moldova, 7
Tags: geometric inequalities , isosceles
In the triangle $ABC$ with $| AB | = c, | BC | = a, | CA | = b$ the relations hold simultaneously $$a \ge max \{ b, c, \sqrt{bc}\}, \sqrt{(a - b) (a + c)} + \sqrt{(a - c) (a + b) } \ge 2\sqrt{a^2-bc}$$ Prove that the triangle $ABC$ is isosceles.

2024 Czech-Polish-Slovak Junior Match, 3
Tags: isosceles , isosceles triangle , geometry
Determine the possible interior angles of isosceles triangles that can be subdivided in two isosceles triangles with disjoint interior.

2013 Greece JBMO TST, 4
Tags: geometry , trapezoid , isosceles
Given the circle $c(O,R)$ (with center $O$ and radius $R$), one diameter $AB$ and midpoint $C$ of the arc $AB$. Consider circle $c_1(K,KO)$, where center $K$ lies on the segment $OA$, and consider the tangents $CD,CO$ from the point $C$ to circle $c_1(K,KO)$. Line $KD$ intersects circle $c(O,R)$ at points $E$ and $Z$ (point $E$ lies on the semicircle that lies also point $C$). Lines $EC$ and $CZ$ intersects $AB$ at points $N$ and $M$ respectively. Prove that quadrilateral $EMZN$ is an isosceles trapezoid, inscribed in a circle whose center lie on circle $c(O,R)$.

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

2004 Dutch Mathematical Olympiad, 4
Tags: geometry , isosceles , tangent circle
Two circles $C_1$ and $C_2$ touch each other externally in a point $P$. At point $C_1$ there is a point $Q$ such that the tangent line in $Q$ at $C_1$ intersects the circle $C_2$ at points $A$ and $B$. The line $QP$ still intersects $C_2$ at point $C$. Prove that triangle $ABC$ is isosceles.

Kyiv City MO Juniors 2003+ geometry, 2004.8.7
Tags: geometry , isosceles , angle
In an isosceles triangle $ABC$ with base $AC$, on side $BC$ is selected point $K$ so that $\angle BAK = 24^o$. On the segment $AK$ the point $M$ is chosen so that $\angle ABM = 90^o$, $AM=2BK$. Find the values ​​of all angles of triangle $ABC$.

2009 District Olympiad, 4
Tags: geometry , isosceles , equilateral , angle
Let $ABC$ be an equilateral $ABC$. Points $M, N, P$ are located on the sides $AC, AB, BC$, respectively, such that $\angle CBM= \frac{1}{2} \angle AMN = \frac{1}{3} \angle BNP$ and $\angle CMP = 90 ^o$. a) Show that $\vartriangle NMB$ is isosceles. b) Determine $\angle CBM$.

2014 Sharygin Geometry Olympiad, 1
Tags: angle chasing , isosceles , geometry , square
The vertices and the circumcenter of an isosceles triangle lie on four different sides of a square. Find the angles of this triangle. (I. Bogdanov, B. Frenkin)

2019 Yasinsky Geometry Olympiad, p2
Tags: geometry , isosceles , centroid , incircle
An isosceles triangle $ABC$ ($AB = AC$) with an incircle of radius $r$ is given. We know that the point $M$ of the intersection of the medians of the triangle $ABC$ lies on this circle. Find the distance from the vertex $A$ to the point of intersection of the bisectrix of the triangle $ABC$. (Grigory Filippovsky)

1963 All Russian Mathematical Olympiad, 040
Tags: geometry , isosceles , locus
Given an isosceles triangle. Find the set of the points inside the triangle such, that the distance from that point to the base equals to the geometric mean of the distances to the sides.

Ukrainian From Tasks to Tasks - geometry, 2012.2
Tags: geometry , locus , equilateral , isosceles
The triangle $ABC$ is equilateral. Find the locus of the points $M$ such that the triangles $ABM$ and $ACM$ are both isosceles.