Found problems: 133
In the triangle $ABC$, the median $AM$ is extended to the intersection with the circumscribed circle at point $D$. It is known that $AB = 2AM$ and $AD = 4AM$. Find the angles of the triangle $ABC$.
(Gryhoriy Filippovskyi)
The medians $AM$ and $BN$ of a triangle $ABC$ are the diameters of the circles $\omega_1$ and $\omega_2$. If $\omega_1$ touches the altitude $CH$, prove that $\omega_2$ also touches $CH$.
I. Gorodnin
On the side $AB$ of the triangle $ABC$ mark the point $K$. The segment $CK$ intersects the median $AM$ at the point $F$. It is known that $AK = AF$. Find the ratio $MF: BK$.
In triangle $ABC$, the altitude and the median from vertex $A$ form (together with line $BC$) a triangle such that the bisectrix of angle $A$ is the median; the altitude and the median from vertex $B$ form (together with line AC) a triangle such that the bisectrix of angle $B$ is the bisectrix. Find the ratio of sides for triangle $ABC$.
In the triangle $ABC$ , the angle bisector $AD$ divides the side $BC$ into the ratio $BD: DC = 2: 1$. In what ratio, does the median $CE$ divide this bisector?
The median $ BM $ of a triangle $ ABC $ intersects the circumscribed circle at point $ K $. The circumcircle of the triangle $ KMC $ intersects the segment $ BC $ at point $ P $, and the circumcircle of $ AMK $ intersects the extension of $ BA $ at $ Q $. Prove that $ PQ> AC $.
In an acute triangle $\vartriangle ABC$, $D$ is the foot of altitude from $A$ to $BC$. Suppose that $AD = CD$, and define $N$ as the intersection of the median $CM$ and the line $AD$. Prove that $\vartriangle ABC$ is isosceles if and only if $CN = 2AM$.
Medians divide the triangle onto $6$ smaller ones. $4$ of the circles inscribed in those small ones are equal. Prove that the triangle is equilateral.
Given a triangle $ABC$ with side $AB$ greater than $BC$, let $M$ be the midpoint of $AC$ and $L$ be the point at which the bisector of angle $\angle B$ intersects side $AC$. The line parallel to $AB$, which intersects the bisector $BL$ at $D$, is drawn by $M$, and the line parallel to the side $BC$ that intersects the median $BM$ at $E$ is drawn by $L$. Show that $ED$ is perpendicular to $BL$.
In the triangle $ABC $ the median $BD$ is drawn, which is divided into three equal parts by the points $E $ and $F$ ($BE = EF = FD$). It is known that $AD = AF$ and $AB = 1$. Find the length of the segment $CE$.
Four lines are given in a plane so that any three of them determine a triangle. One of these lines is parallel to a median in the triangle determined by the other three lines. Prove that each of the other three lines also has this property.
Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$.
[i]Proposed by Evan Chen, Taiwan[/i]
Lines $b$ and $c$ passing through vertices $B$ and $C$ of triangle $ABC$ are perpendicular to sideline $BC$. The perpendicular bisectors to $AC$ and $AB$ meet $b$ and $c$ at points $P$ and $Q$ respectively. Prove that line $PQ$ is perpendicular to median $AM$ of triangle $ABC$.
(D. Prokopenko)
In a triangle $ABC$, the median from $B$ to $CA$ is perpendicular to the median from $C$ to $AB$.
If the median from $A$ to $BC$ is $30$, determine $\frac{BC^2 + CA^2 + AB^2}{100}$.
The angles of an acute triangle $ABC$ at $\alpha , \beta, \gamma$. Let $AD$ be a height, $CF$ a median, and $BE$ the bisector of $\angle B$. Show that $AD,CF$ and $BE$ are concurrent if and only if $\cos \gamma \tan\beta = \sin \alpha$ .
On sides $BC$, $CA$, $AB$ of triangle $ABC$, points $A_1$, $B_1$, $C_1$ are chosen, respectively, so that the medians $A_1A_2$, $B_1B_2$, $C_1C_2$ of the triangle $A_1B_1C_1$ are respectively parallel to straight lines $AB$, $BC$, $CA$. Determine in what ratio points $A_1$, $B_1$, $C_1$ divide the sides of the triangle $ABC$.
Let $ABC$ be any triangle and $D, E, F$ the midpoints of $BC, CA, AB$. The medians $AD, BE$ and $CF$ intersect at point $S$. At least two of the quadrilaterals $AF SE, BDSF, CESD$ are cyclic. Show that the triangle $ABC$ is equilateral.
Let $ABC$ be any triangle and $D, E, F$ the midpoints of $BC, CA, AB$. The medians $AD, BE$ and $CF$ intersect at point $S$. At least two of the quadrilaterals $AF SE, BDSF, CESD$ are cyclic. Show that the triangle $ABC$ is equilateral.
What is the median of the following list of $4040$ numbers$?$
$$1, 2, 3, ..., 2020, 1^2, 2^2, 3^2, ..., 2020^2$$
$\textbf{(A) } 1974.5 \qquad \textbf{(B) } 1975.5 \qquad \textbf{(C) } 1976.5 \qquad \textbf{(D) } 1977.5 \qquad \textbf{(E) } 1978.5$
In a triangle $ABC$, points $H,M,L$ are the feet of the altitude from $C$, the median from $A$, and the angle bisector from $B$, respectively. Show that if triangle $HML$ is equilateral, then so is triangle $ABC$.
In triangle $ABC$, $BD$ is a median. $CF$ intersects $BD$ at $E$ so that $\overline{BE}=\overline{ED}$. Point $F$ is on $AB$. Then, if $\overline{BF}=5$, $\overline{BA}$ equals:
$ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ \text{none of these} $
Numbers $a, b, c$ are the length of the medians of some triangle. If $ab + bc + ac = 1$ prove that
a) $a^2b + b^2c + c^2a > \frac13$ b) $a^2b + b^2c + c^2a > \frac12$
(I. Bliznets)
Find all real-valued functions $f$ defined on pairs of real numbers, having the following property: for all real numbers $a, b, c$, the median of $f(a,b), f(b,c), f(c,a)$ equals the median of $a, b, c$.
(The [i]median[/i] of three real numbers, not necessarily distinct, is the number that is in the middle when the three numbers are arranged in nondecreasing order.)
The medians $AL, BM$, and $CN$ are drawn in the triangle $ABC$. Prove that $\angle ANC = \angle ALB$ if and only if $\angle ABM =\angle LAC$.
(Veklich Bogdan)
Let $AH$, $BL$ and $CM$ be an altitude, a bisectrix and a median in triangle $ABC$. It is known that lines $AH$ and $BL$ are an altitude and a bisectrix of triangle $HLM$. Prove that line $CM$ is a median of this triangle.