Found problems: 133
Let \( AL \) be the bisector of triangle \( ABC \), \( O \) the center of its circumcircle, and \( D \) and \( E \) the midpoints of \( BL \) and \( CL \), respectively. Points \( P \) and \( Q \) are chosen on segments \( AD \) and \( AE \) such that \( APLQ \) is a parallelogram. Prove that \( PQ \perp AO \).
[i]Proposed by Mykhailo Plotnikov[/i]
Suppose medians $m_a$ and $m_b$ of a triangle are orthogonal. Prove that:
a.) Using medians of that triangle it is possible to construct a rectangular triangle.
b.) The following inequality: \[5(a^2+b^2-c^2) \geq 8ab,\] is valid, where $a,b$ and $c$ are side length of the given triangle.
Construct a triangle $ABC$ with the right angle at vertex $C$ given lengths of its medians $m_a$, $m_b$. Discuss conditions of solvability.
Median $AM$ and the angle bisector $CD$ of a right triangle $ABC$ ($\angle B=90^o$) intersect at the point $O$. Find the area of the triangle $ABC$ if $CO=9, OD=5$.
In a triangle $ABC$, right angled at $A$, the altitude through $A$ and the internal bisector of $\angle A$ have lengths $3$ and $4$, respectively. Find the length of the median through $A$.
Circles $k_1$ and $k_2$ intersect in points $A$ and $B$, such that $k_1$ passes through the center $O$ of the circle $k_2$. The line $p$ intersects $k_1$ in points $K$ and $O$ and $k_2$ in points $L$ and $M$, such that the point $L$ is between $K$ and $O$. The point $P$ is orthogonal projection of the point $L$ to the line $AB$. Prove that the line $KP$ is parallel to the $M-$median of the triangle $ABM$.
[i]Matko Ljulj[/i]
Let $ A_1,B_1,C_1 $ be the middlepoints of the sides of a triangle $ ABC $ and let $ A_2,B_2,C_2 $ be on the middle of the paths $ CAB,ABC,BCA, $ respectively. Prove that $ A_1A_2,B_1B_2,C_1C_2 $ are concurrent.
In a triangle $ABC$ the area is $18$, the length $AB$ is $5$, and the medians from $A$ and $B$ are orthogonal. Find the lengths of the sides $BC,AC$.
For the given triangle $ABC$, prove that a point $X$ on the side $AB$ satisfies the condition $\overrightarrow{XA} \cdot\overrightarrow{XB} +\overrightarrow{XC} \cdot \overrightarrow{XC} = \overrightarrow{CA} \cdot \overrightarrow{CB} $, iff $X$ is the basepoint of the altitude or median of the triangle $ABC$.
Demonstrate that the condition necessary so that, in triangle ${ABC}$, the median from ${B}$ is divided into three equal parts by the inscribed circumference of a circle is:
${A/5 = B/10 = C/13}$.
Let $ABC$ be a triangle with median AK. Let $O$ be the circumcenter of the triangle $ABK$.
a) Prove that if $O$ lies on a midline of the triangle $ABC$, but does not coincide with its endpoints, then $ABC$ is a right triangle.
b) Is the statement still true if $O$ can coincide with an endpoint of the midsegment?
Let $BE$ and $CF$ be the medians of an acute triangle $ABC.$ On the line $BC,$ points $K \ne B$ and $L \ne C$ are chosen such that $BE = EK$ and $CF = FL.$ Prove that $AK = AL.$
[i]Proposed by Heorhii Zhilinskyi[/i]
In the triangle $ABC$, the medians from $B$ and $C$ are perpendicular. Show that $\cot B + \cot C \ge \frac23$.
The median $CM$ of the triangle $ABC$ is equal to the bisector $BL$, also $\angle BAC=2\angle ACM$. prove that the triangle is right.
Construct a right triangle with given hypotenuse $c$ such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.
Let $BM$ be the median of the triangle $ABC$, in which $AB> BC$. Point $P$ is chosen so that $AB \parallel PC$ and$ PM \perp BM$. The point $Q$ is chosen on the line $BP$ so that $\angle AQC = 90^o$, and the points $B$ and $Q$ lie on opposite sides of the line $AC$. Prove that $AB = BQ$.
(Mikhail Standenko)
It is given that for no side of the triangle from the height drawn to it, the bisector and the median it is impossible to make a triangle. Prove that one of the angles of the triangle is greater than $135^o$
Medians $AA'$ and $BB'$ of triangle $ABC$ meet at point $M$, and $\angle AMB = 120^o$.
Prove that angles $AB'M$ and $BA'M$ are neither both acute nor both obtuse.
A median , a bisector and an altitude of a certain triangle intersect at an inner point $O$ . The segment of the bisector from the vertex to $O$ is equal to the segment of the altitude from the vertex to $O$ . Prove that the triangle is equilateral .
The median $BM$ is drawn in the triangle $ABC$. It is known that $\angle ABM = 40 {} ^ \circ$ and $\angle CBM = 70 {} ^ \circ $ Find the ratio $AB: BM$.
Does there exist a nonisosceles triangle such that the altitude from one vertex, the bisectrix from the second one and the median from the third one are equal?
Let $A B C$ be a triangle in the $x y$ plane, where $B$ is at the origin $(0,0)$. Let $B C$ be produced to $D$ such that $B C: C D=1: 1, C A$ be produced to $E$ such that $C A: A E=1: 2$ and $A B$ be produced to $F$ such that $A B: B F=1: 3$. Let $G(32,24)$ be the centroid of the triangle $A B C$ and $K$ be the centroid of the triangle $D E F$. Find the length $G K$.
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 $ABC$.
The ratio of the median $AM$ of a triangle $ABC$ to the side $BC$ equals $\sqrt{3}:2$. The points on the sides of $ABC$ dividing these side into $3$ equal parts are marked. Prove that some $4$ of these $6$ points are concyclic.
Medians $AM$ and $BE$ of a triangle $ABC$ intersect at $O$. The points $O, M, E, C$ lie on one circle. Find the length of $AB$ if $BE = AM =3$.