Found problems: 333
Let $ABC$ be a right triangle with $\angle ABC = 90^o$ . Points $D$ and $E$, located on the legs $(AC)$ and $(AB)$ respectively, are the legs of the inner bisectors taken from the vertices $B$ and $C$, respectively. Let $I$ be the center of the circle inscribed in the triangle $ABC$. If $BD \cdot CE = m^2 \sqrt2$ , find the area of the triangle $BIC$ (in terms of parameter $m$)
The student drew a right triangle $ABC$ on the board with a right angle at the vertex $B$ and inscribed in it an equilateral triangle $KMP$ such that the points $K, M, P$ lie on the sides $AB, BC, AC$, respectively, and $KM \parallel AC$. Then the picture was erased, leaving only points $A, P$ and $C$. Restore erased points and lines.
Two circles $\Gamma_a$ and $\Gamma_b$ with their centres lying on the legs $BC$ and $CA$ of a right triangle, both touching the hypotenuse $AB$, and both passing through the vertex $C$ are given. Let the radii of these circles be denoted by $\gamma_a$ and $\gamma_b$.
Find the greatest real number $p$ such that the inequality $\frac{1}{\gamma_a}+\frac{1}{\gamma_b}\ge p \left(\frac{1}{a}+\frac{1}{b}\right)$
($BC = a,CA = b$) holds for all right triangles $ABC$.
Let ABC be a triangle with $\angle A = 90^o$, $\angle B = 75^o$, and $AB = 2$. Points $P$ and $Q$ of the sides $AC$ and $BC$ respectively, are such that $\angle APB = \angle CPQ$ and $\angle BQA = \angle CQP$. Calculate the lenght of $QA$.
Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\overline{AB}$, and for $n\geq 2$, let $C_n$ be the foot of the altitude to $\overline{C_{n-2}B}$ in $\triangle C_{n-2}C_{n-1}B$. The sum $\sum\limits_{n=1}^{\infty}C_{n-1}C_n = 6p$. Find $p$.
In the right triangle $ABC$ ( $\angle A = 90^o$), $D$ is the intersection of the bisector of the angle $A$ with the side $(BC)$, and $P$ and $Q$ are the projections of the point $D$ on the sides $(AB),(AC)$ respectively . If $BQ \cap DP=\{M\}$, $CP \cap DQ=\{N\}$, $BQ\cap CP=\{H\}$, show that:
a) $PM = DN$
b) $MN \parallel BC$
c) $AH \perp BC$.
In a right triangle $ ABC $, the altitude $ CD $ is drawn from the vertex of the right angle $ C $ and a circle is inscribed in each of the triangles $ ABC $, $ ACD $ and $ BCD $. Prove that the sum of the radii of these circles equals the height $ CD $.
Show that in a right-angled triangle the bisector of the right angle divides into equal parts the angle between the altitude and the median, drawn from the same vertex.
Let $O$ be the center of the circle circumscribed around the the triangle $ABC$. The centers of the circles circumscribed around the squares $OAB,OBC,OCA$ lie at the vertices of a regular triangle. Prove that the triangle $ABC$ is right.
Determine for what values of $x$ the expressions $2x + 2$,$x + 4$, $x + 2$ can represent the sidelengths of a right triangle.
Prove that a right triangle has an angle equal to $30^o$ if and only if the center of the circle inscribed in this triangle is located on the perpendicular bisector of the median taken from the vertex of the right angle of the triangle.
Points $E$ and $F$ are chosen on the sides $AB$ and $BC$ respectively of a square $ABCD$ such that $BE=BF$. Let $BN$ be an altitude of the triangle $BCE$. Prove that the triangle $DNF$ is right-angled.
Let $ABC$ be a triangle with $\angle C = 90^o$ and $AC = 1$. The median $AM$ intersects the incircle at the points $P$ and $Q$, with $P$ between $A$ and $Q$, such that $AP = QM$. Find the length of $PQ$.
Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$.
Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.
Triangle $ABC$ with $AB = 10$ cm ¸and $\angle C= 15^o$, is right at $B$. Point $D \in (AC)$ is the foot of the altitude taken from $B$. Find the distance from point $D$ to the line $AB$.
In triangle $ABC$, angle $C$ is right and the two catheti are both length $1$. For one given the choice of the point $P$ on the cathetus $BC$, the point $Q$ on the hypotenuse and the point $R$ are plotted on the second cathetus so that $PQ$ is parallel to $AC$ and $QR$ is parallel to $BC$. Thereby the triangle is divided into three parts. Determine the locations of point $P$ for which the rectangular part has a larger area than each of the other two parts.
In the right triangle $ABC$ ($m ( \angle A) = 90^o$) $D$ is the foot of the altitude from $A$. The bisectors of the angles $ABD$ and $ADB$ intersect in $I_1$ and the bisectors of the angles $ACD$ and $ADC$ in $I_2$. Find the angles of the triangle if the sum of distances from $I_1$ and $I_2$ to $AD$ is equal to $\frac14$ of the length of $BC$.
Show that in a right-angled triangle the bisector of the right angle divides into equal parts the angle between the altitude and the median, drawn from the same vertex.
Consider a triangle such that \[\sin^2\alpha+\sin^2\beta+\sin^2\gamma=2.\] Show that the triangle is right.
Let $ABC$ be a triangle, right-angled at point $ A$ and with $AB>AC$. The tangent through $ A$ of the circumcircle $G$ of $ABC$ cuts $BC$ at $D$. $E$ is the reflection of $ A$ over line $BC$. $X$ is the foot of the perpendicular from $ A$ over $BE$. $Y$ is the midpoint of $AX$, $Z$ is the intersection of $BY$ and $G$ other than $ B$, and $F$ is the intersection of $AE$ and $BC$. Prove $D, Z, F, E$ are concyclic.
Consider the triangles $[ABC]$ and $[EDC]$, right at $A$ and $D$, respectively. Show that, if $E$ is the midpoint of $[AC]$, then $AB <BD$.
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A right triangle is drawn on the plane. How to use only a compass to mark two points, such that the distance between them is equal to the diameter of the circle inscribed in this triangle?
We are given a right triangle $ABC$ and the median $BD$ drawn from the vertex $B$ of the right angle. Let the circle inscribed in $\vartriangle ABD$ be tangent to side $AD$ at $K$. Find the angles of $\vartriangle ABC$ if $K$ divides $AD$ in halves.
In a right-angled triangle in which all side lengths are integers, one has a cathetus length $1994$. Determine the length of the hypotenuse.
On the hypotenuse $AB$ of a right triangle $ABC$, we denote a point $K$ such that $BK = BC$. Let $P$ be a point on the perpendicular from the point $K$ to line $CK$, equidistant from the points $K$ and $B$. Let $L$ be the midpoint of $CK$. Prove that line $AP$ is tangent to the circumcircle of $\Delta BLP$.