Found problems: 333
Let $ABC$ be an isosceles right triangle with $\angle BCA=90^{\circ}, BC=AC=10$. Let $P$ be a point on $AB$ that is a distance $x$ from $A$, $Q$ be a point on $AC$ such that $PQ$ is parallel to $BC$. Let $R$ and $S$ be points on $BC$ such that $QR$ is parallel to $AB$ and $PS$ is parallel to $AC$. The union of the quadrilaterals $PBRQ$ and $PSCQ$ determine a shaded area $f(x)$. Evaluate $f(2)$
In the figure, $[ABC]$ and $[DEC]$ are right triangles . Knowing that $EB = 1/2, EC = 1$ and $AD = 1$, calculate $DC$.
[img]https://1.bp.blogspot.com/-nAOrVnK5JmI/X4UMb2CNTyI/AAAAAAAAMmk/TtaBESxYyJ0FsBoY2XaCGlCTc6mgmA5TQCLcBGAsYHQ/s0/2000%2Bportugal%2Bp5.png[/img]
In triangle $ABC$ we have $\angle C = 90^o$ and $AC = BC$. Furthermore $M$ is an interior pont in the triangle so that $MC = 1 , MA = 2$ and $MB =\sqrt2$. Determine $AB$
In the triangle $ABC, AC=BC, \angle C=90^o, D$ is the midpoint of $BC, E$ is the point on $AB$ such that $AD$ is perpendicular to $CE$. Prove that $AE=2EB$.
The incircle of triangle $ABC$ touches its sides in points $A', B',C'$ . It is known that the orthocenters of triangles $ABC$ and $A' B'C'$ coincide. Is triangle $ABC$ regular?
In a right-angled triangle in which all side lengths are integers, one has a cathetus length $1994$. Determine the length of the hypotenuse.
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 $\angle ACB = 90^{\circ}$.
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 $\angle ACB = 90^{\circ}$.
The catheti $AC$ and $BC$ in a right-angled triangle $ABC$ have lengths $b$ and $a$, respectively. A circle centered at $C$ is tangent to hypotenuse $AB$ at point $D$. The tangents to the circle through points $A$ and $B$ intersect the circle at points $E$ and $F$, respectively (where $E$ and $F$ are both different from $D$). Express the length of the segment $EF$ in terms of $a$ and $b$.
In the figure below the triangles $BCD, CAE$ and $ABF$ are equilateral, and the triangle $ABC$ is right-angled with $\angle A = 90^o$. Prove that $|AD| = |EF|$.
[img]https://1.bp.blogspot.com/-QMMhRdej1x8/XzP18QbsXOI/AAAAAAAAMUI/n53OsE8rwZcjB_zpKUXWXq6bg3o8GUfSwCLcBGAsYHQ/s0/2019%2Bmohr%2Bp5.png[/img]
The trapezoid $ABCD$ of bases $AB$ and $CD$, has $\angle A = 90^o, AB = 6, CD = 3$ and $AD = 4$. Let $E, G, H$ be the circumcenters of triangles $ABC, ACD, ABD$, respectively. Find the area of the triangle $EGH$.
In an isosceles right triangle $ABC$ the right angle is at vertex $C$. On the side $AC$ points $K, L$ and on the side $BC$ points $M, N$ are chosen so that they divide the corresponding side into three equal segments. Prove that there is exactly one point $P$ inside the triangle $ABC$ such that $\angle KPL = \angle MPN = 45^o$.
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.
The bisectors $AA_1$ and $CC_1$ of the right triangle $ABC$ ($\angle B = 90^o$) intersect at point $I$. The line passing through the point $C_1$ and perpendicular on the line $AA_1$ intersects the line that passes through $A_1$ and is perpendicular on $CC_1$, at the point $K$. Prove that the midpoint of the segment $KI$ lies on segment $AC$.
On the legs $AC$ and $BC$ of an isosceles right-angled triangle with a right angle $C$, points $D$ and $E$ are taken, respectively, so that $CD = CE$. Perpendiculars on line $AE$ from points $C$ and $D$ intersect segment $AB$ at points $P$ and $Q$, respectively. Prove that $BP = PQ$.
Eight circles of radius $r$ located in a right triangle $ABC$ (angle $C$ is right) as shown in figure (each of the circles touches the respactive sides of the triangle and the other circles). Find the radius of the inscribed circle of triangle $ABC$.
[img]https://cdn.artofproblemsolving.com/attachments/4/7/1b1cd7d6bc7f5004b8e94468d723ed16e9a039.png[/img]
Let $ABC$ be an isosceles right-angled triangle, having the right angle at vertex $C$. Let us consider the line through $C$ which is parallel to $AB$ and let $D$ be a point on this line such that $AB = BD$ and $D$ is closer to $B$ than to $A$. Find the angle $\angle CBD$.
Points $M$ and $N$ are taken on the hypotenuse $AB$ of a right triangle $ABC$ so that $BC = BM$ and $AC = AN$. Prove that the angle $MCN$ is equal to $45$ degrees.
(Folklore)
In a triangle $ABC$ with $\angle BC A = 90^o$, the perpendicular bisector of $AB$ intersects segments $AB$ and $AC$ at $X$ and $Y$, respectively. If the ratio of the area of quadrilateral $BXYC$ to the area of triangle $ABC$ is $13 : 18$ and $BC = 12$ then what is the length of $AC$?
In a right-angled triangle $ ABC$ let $ AD$ be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles $ ABD, ACD$ intersect the sides $ AB, AC$ at the points $ K,L$ respectively. If $ E$ and $ E_1$ dnote the areas of triangles $ ABC$ and $ AKL$ respectively, show that
\[ \frac {E}{E_1} \geq 2.
\]
Outside the square $ABCD$, the rhombus $BCMN$ is constructed with angle $BCM$ obtuse . Let $P$ be the intersection point of the lines $BM$ and $AN$ . Prove that $DM \perp CP$ and the triangle $DPM$ is right isosceles .
The sidelengths of a triangle and the diameter of its incircle, taken in some order, form an arithmetic progression. Prove that the triangle is right-angled.
Three right-angled triangles have been placed in a halfplane determined by a line $\ell$, each with one leg lying on $\ell$. Assume that there is a line parallel to $\ell$ cutting the triangles in three congruent segments. Show that, if each of the triangles is rotated so that its other leg lies on $\ell$, then there still exists a line parallel to $\ell$ cutting them in three congruent segments.
Let $ABC$ be a right triangle with $\angle ACB = 90^{\circ}$ and centroid $G$. The circumcircle $k_1$ of triangle $AGC$ and the circumcircle $k_2$ of triangle $BGC$ intersect $AB$ at $P$ and $Q$, respectively. The perpendiculars from $P$ and $Q$ respectively to $AC$ and $BC$ intersect $k_1$ and $k_2$ at $X$ and $Y$. Determine the value of $\frac{CX \cdot CY}{AB^2}$.
Given triangle $ABC$. Lines $AL_a$ and $AM_a$ are the internal and the external bisectrix of angle $A$. Let $\omega_a$ be the reflection of the circumcircle of $\triangle AL_aM_a$ in the midpoint of $BC$. Circle $\omega_b$ is defined similarly. Prove that $\omega_a$ and $\omega_b$ touch if and only if $\triangle ABC$ is right-angled.