Found problems: 333
Let $ABC$ be a right triangle and isosceles, with $\angle C = 90^o$. Let $M$ be the midpoint of $AB$ and $N$ the midpoint of $AC$. Let $ P$ be such that $MNP$ is an equilateral triangle with $ P$ inside the quadrilateral $MBCN$. Calculate the measure of $\angle CAP$
The height drawn on the hypotenuse of a right triangle divides the hypotenuse into two sections with a length ratio of $9: 1$ and two triangles of the starting triangle with a difference of areas of $48$ cm$^2$. Find the original triangle sidelengths.
In triangle $ABC$, $AB = AC = 3$ and $\angle A = 90^o$. Let $M$ be the midpoint of side $BC$. Points $D$ and $E$ lie on sides $AC$ and $AB$ respectively such that $AD > AE$ and $ADME$ is a cyclic quadrilateral. Given that triangle $EMD$ has area $2$, find the length of segment $CD$.
An isosceles right-angled triangle $ABC$ is given. On the extensions of sides $AB$ and $AC$, behind vertices $B$ and $C$ equal segments $BK$ and $CL$ were laid. $E$ and F are the points of intersection of the segment $KL$ and the lines perpendicular to the $KC$ , passing through the points $B$ and $A$, respectively. Prove that $EF = FL$.
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 $\vartriangle ABC$ be a right-angled triangle with $\angle A = 90^o$ and circumcircle $\Gamma$. The inscribed circle is tangent to $BC$ in point $D$. Let $E$ be the midpoint of the arc $AB$ of $\Gamma$ not containing $C$ and let $F$ be the midpoint of the arc $AC$ of $\Gamma$ not containing $B$.
(a) Prove that $\vartriangle ABC \sim \vartriangle DEF$.
(b) Prove that $EF$ goes through the points of tangency of the incircle to $AB$ and $AC$.
Construct a right-angled triangle along its two medians, starting from the acute angles.
Show that in any triangle $ABC$ with $A = 90^0$ the following inequality holds:
$$(AB -AC)^2(BC^2 + 4AB \cdot AC)^2 \le 2BC^6$$
Two right triangles are given, of which the incircle of the first triangle is the circumcircle of the second triangle. Let the areas of the triangles be $S$ and $S'$ respectively. Prove that $\frac{S}{S'} \ge 3 +2\sqrt2$
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 $\angle ACB = 90^{\circ}$.
A triangle $\vartriangle ABC$ has $\angle B = 90^o$ . A circle is tangent to $AB$ at $B$ and also tangent to $AC$. Another circle is tangent to the first circle as well as the two sides $AB$ and $AC$. Suppose that $AB =\sqrt3$ and $BC = 3$. What is the radius of the second circle?
Let $ABC$ be a right isosceles triangle with right angle at $A$. Let $E$ and $F$ be points on A$B$ and $AC$ respectively such that $\angle ECB = 30^o$ and $\angle FBC = 15^o$. Lines $CE$ and $BF$ intersect at $P$ and line $AP$ intersects side $BC$ at $D$. Calculate the measure of angle $\angle FDC$.
In a right-angled and isosceles triangle, the two catheti are both length $1$. Find the length of the shortest line segment dividing the triangle into two figures with the same area, and specify the location of this line segment
Let $D$ be the midpoint of the hypotenuse $AB$ of a right triangle $ABC$. Let $O_1$ and $O_2$ be the circumcenters of the $ADC$ and $DBC$ triangles, respectively.
a) Prove that $\angle O_1DO_2$ is right.
b) Prove that $AB$ is tangent to the circle of diameter $O_1O_2$ .
Let $ABC$ be a right triangle in $A$ , whose leg measures $1$ cm. The bisector of the angle $BAC$ cuts the hypotenuse in $R$, the perpendicular to $AR$ on $R$ , cuts the side $AB$ at its midpoint. Find the measurement of the side $AB$ .
Let $ABC$ be a right triangle with leg $CB = 2$ and hypotenuse $AB= 4$. Point $K$ is chosen on the hypotenuse $AB$, and point $L$ is chosen on the leg $AC$.
a) Describe and justify how to construct such points $K$ and $ L$ so that the sum of the distances $CK+KL$ is the smallest possible.
b) Find the smallest possible value of $CK+KL$.
(Olexii Panasenko)
A triangle has sides $a, b, c$ with longest side $c$, and circumradius $R$. Show that if $a^2 + b^2 = 2cR$, then the triangle is right-angled.
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 $\angle ACB = 90^{\circ}$.
Determine whether there is a tetrahedron $ABCD$ with the longest edge of length 1 such that all its faces are similar right triangles with right angles at vertices $B,C.$ If so, determine which edge is the longest, which is the shortest and what is its length.
A right-angled triangle has property that, when a square is drawn externally on each side of the triangle, the six vertices of the squares that are not vertices of the triangle are concyclic. Assume that the area of the triangle is $9$ cm$^2$. Determine the length of sides of the triangle.
In $\vartriangle ABC, \angle ACB = 90^o, D$ is the foot of the altitude from $C$ to $AB$ and $E$ is the point on the side $BC$ such that $CE = BD/2$. Prove that $AD + CE = AE$.
In a right triangle, the altitude from a vertex to the hypotenuse splits the hypotenuse into two segments of lengths $a$ and $b$. If the right triangle has area $T$ and is inscribed in a circle of area $C$, find $ab$ in terms of $T$ and $C$.
Let $ABC$ be a triangle right in $A$. Construct a point $P$ on the hypotenuse $BC$ such that if $Q$ is the foot of the perpendicular drawn from $P$ to side $AC$, then the area of the square of side $PQ$ is equal to the area of the rectangle of sides $PB$ and $PC$. Show construction steps.
A right triangle $ABC$ has the right angle at vertex $A$. Circle $c$ passes through vertices $A$ and $B$ of the triangle $ABC$ and intersects the sides $AC$ and $BC$ correspondingly at points $D$ and $E$. The line segment $CD$ has the same length as the diameter of the circle $c$. Prove that the triangle $ABE$ is isosceles.
The right triangles $ABC$ and $AB_1C_1$ are similar and have opposite orientation. The right angles are at $C$ and $C_1$, and we also have $ \angle CAB = \angle C_1AB_1$. Let $M$ be the point of intersection of the lines $BC_1$ and $B_1C$. Prove that if the lines $AM$ and $CC_1$ exist, they are perpendicular.