Found problems: 351
In acute triangle $ABC$, $\angle A = 20^o$. Prove that the triangle is isosceles if and only if $$\sqrt[3]{a^3 + b^3 + c^3 -3abc} = \min\{b, c\}$$, where $a,b, c$ are the side lengths of triangle $ABC$.
A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all others are equilateral. Determine the angles of the original triangle.
Let \( ABC \) be an acute-angled scalene triangle with circumcenter \( O \). Denote by \( M \), \( N \), and \( P \) the midpoints of sides \( BC \), \( CA \), and \( AB \), respectively. Let \( \omega \) be the circle passing through \( A \) and tangent to \( OM \) at \( O \). The circle \( \omega \) intersects \( AB \) and \( AC \) at points \( E \) and \( F \), respectively (where \( E \) and \( F \) are distinct from \( A \)). Let \( I \) be the midpoint of segment \( EF \), and let \( K \) be the intersection of lines \( EF \) and \( NP \). Prove that \( AO = 2IK \) and that triangle \( IMO \) is isosceles.
Point $O$ is the center of circumcircle $\omega$ of the isosceles triangle $ABC$ ($AB = AC$). Bisector of the angle $\angle C$ intersects $\omega$ at the point $W$. Point $Q$ is the center of the circumcircle of the triangle $OWB$. Construct the triangle $ABC$ given the points $Q,W, B$.
(Andrey Mostovy)
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
Given a triangle $ABC$, in which $AB = BC$. Point $O$ is the center of the circumcircle, point $I$ is the center of the incircle. Point $D$ lies on the side $BC$, such that the lines $DI$ and $AB$ parallel. Prove that the lines $DO$ and $CI$ are perpendicular.
(Vyacheslav Yasinsky)
A trapezoid and an isosceles triangle are inscribed in a circle. The larger base of the trapezoid is the diameter of the circle, and the sides of the triangle are parallel to the sides of the trapezoid. Show that the trapezoid and the triangle have equal areas.
Let $d$ be the tangent at $B$ to the circumcircle of the acute scalene triangle $ABC$. Let $K$ be the orthogonal projection of the orthocenter, $H$, of triangle $ABC$ to the line $d$ and $L$ the midpoint of the side $AC$. Prove that the triangle $BKL$ is isosceles.
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.
In $\vartriangle ABC, D$ and $E$ are two points on segment $BC$ such that $BD = CE$ and $\angle BAD = \angle CAE$. Prove that $\vartriangle ABC$ is isosceles
Consider $\triangle ABC$ an isosceles triangle such that $AB = BC$. Let $P$ be a point satisfying
$$\angle ABP = 80^\circ, \angle CBP = 20^\circ, \textrm{and} \hspace{0.17cm} AC = BP$$
Find all possible values of $\angle BCP$.
In triangle $[ABC]$, the bisector in $C$ and the altitude passing through $B$ intersect at point $D$. Point $E$ is the symmetric of point $D$ wrt $BC$ and lies on the circle circumscribed to the triangle $[ABC]$. Prove that the triangle is $[ABC]$ isosceles.
Cut a right triangle with an angle of $30^o$ into three isosceles non-acute triangles, among which there are no congruent ones.
(Maria Rozhkova)
Let $ABCD$ be a convex quadrilateral. Suppose that similar isosceles triangles $APB, BQC, CRD, DSA$ with the bases on the sides of $ABCD$ are constructed in the exterior of the quadrilateral such that $PQRS$ is a rectangle but not a square. Show that $ABCD$ is a rhombus.
An isosceles trapezoid $ABCD$ ($AB = CD$) is given. A point $P$ on its circumcircle is such that segments $CP$ and $AD$ meet at point $Q$. Let $L$ be tha midpoint of$ QD$. Prove that the diagonal of the trapezoid is not greater than the sum of distances from the midpoints of the lateral sides to ana arbitrary point of line $PL$.
An equilateral triangle $DCE$ is placed outside a square $ABCD$. The center of this triangle is denoted as $M$ and the intersection of the straight line $AC$ and $BE$ with $S$. Prove that the triangle $CMS$ is isosceles.
Bisectrices $AA_1$ and $BB_1$ of triangle $ABC$ meet in $I$. Segments $A_1I$ and $B_1I$ are the bases of isosceles triangles with opposite vertices $A_2$ and $B_2$ lying on line $AB$. It is known that line $CI$ bisects segment $A_2B_2$. Is it true that triangle $ABC$ is isosceles?
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.
An angle with vertex $A$ is given on the plane. Points $K$ and $P$ are selected on its sides so that $AK + AP = a$, where $a$ is a given segment. Let $M$ be a point on the plane such that the triangle $KPM$ is isosceles with the base $KP$ and the angle at the vertex $M$ equal to the given angle. Find the locus of points $M$ (as $K$ and $P$ move).
Given an isosceles triangle $ABC$ with a vertex at the point $B$. Based on $AC$, an arbitrary point $D $ is selected, different from the vertices $A$ and $C $. On the line $AC $ select the point $E $ outside the segment $AC$, for which $AE = CD$. Prove that the perimeter $\Delta BDE$ is larger than the perimeter $\Delta ABC$.
Each diagonal of a quadrangle divides it into two isosceles triangles. Is it true that the quadrangle is a diamond?
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.
All rectangles that can be inscribed in an isosceles triangle with two of their vertices on the triangle’s base have the same perimeter. Construct the triangle.
In the triangle $ABC$, the bisector of angle $\angle B$ meets $AC$ at $D$ and the bisector of angle $\angle C$ meets $AB$ at $E$. The bisectors meet each other at $O$. Furthermore, $OD = OE$. Prove that either $ABC$ is isosceles or $\angle BAC = 60^\circ$.
On the base $BC$ of the isosceles triangle $ABC$ chose a point $D$ and in each of the triangles $ABD$ and $ACD$ inscribe a circle. Then everything was wiped, leaving only two circles. It is known from which side of their line of centers
the apex $A$ is located . Use a compass and ruler to restore the triangle $ABC$ , if we know that :
a) $AD$ is angle bisector,
b) $AD$ is median.