Found problems: 333
A right triangle when rotating around a large leg forms a cone with a volume of $100\pi$. Calculate the length of the path that passes through each vertex of the triangle at rotation of $180^o$ around the point of intersection of its bisectors, if the sum of the diameters of the circles, inscribed in the triangle and circumscribed around it, are equal to $17$.
In a right triangle rectangle $ABC$ such that $AB = AC$, $M$ is the midpoint of $BC$. Let $P$ be a point on the perpendicular bisector of $AC$, lying in the semi-plane determined by $BC$ that does not contain $A$. Lines $CP$ and $AM$ intersect at $Q$. Calculate the angles that form the lines $AP$ and $BQ$.
A right-angled triangle has perimeter $60$ and the altitude of the hypotenuse has a length $12$. Determine the lengths of the sides.
Let $ABC$ be an isosceles right triangle at $A$ with $AB=AC$. Let $M$ and $N$ be on side $BC$, with $M$ between $B$ and $N,$ such that $$BM^2+ NC^2= MN^2.$$ Determine the measure of the angle $\angle MAN.$
Let $ABC$ be an isosceles right-angled triangle. Point $D$ is chosen on the prolongation of the hypothenuse $AB$ beyond point $A$ so that $AB = 2AD$. Points $M$ and $N$ on side $AC$ satisfy the relation $AM = NC$. Point $K$ is chosen on the prolongation of $CB$ beyond point $B$ so that $CN = BK$. Determine the angle between lines $NK$ and $DM$.
(M.Kungozhin)
Let $CH$ be the altitude of the triangle $ABC$ drawn on the board, in which $\angle C = 90^o$, $CA \ne CB$. The mathematics teacher drew the perpendicular bisectors of segments$ CA$ and $CB$, which cut the line CH at points $K$ and $M$, respectively, and then erased the drawing, leaving only the points $C, K$ and $M$ on the board. Restore triangle $ABC$, using only a compass and a ruler.
Determine the locus of points that are the midpoints of segments of equal length, the ends of which lie on the sides of a given right angle.
Triangle $ABC$ has side lengths $AB = 3$, $BC = 4$, and $CD = 5$. Draw line $\ell_A$ such that $\ell_A$ is parallel to $BC$ and splits the triangle into two polygons of equal area. Define lines $\ell_B$ and $\ell_C$ analogously. The intersection points of $\ell_A$, $\ell_B$, and $\ell_C$ form a triangle. Determine its area.
It is known that in a right triangle:
a) The height drawn from the top of the right angle is the geometric mean of the projections of the legs on the hypotenuse;
b) the leg is the geometric mean of the hypotenuse and the projection of this leg to the hypotenuse.
Are the converse statements true? Formulate them and justify the answer.
Is it possible to formulate the criterion of a right triangle based on these statements? If possible, then how? If not, why?
An altitude $AH$ of triangle $ABC$ bisects a median $BM$. Prove that the medians of triangle $ABM$ are sidelengths of a right-angled triangle.
by Yu.Blinkov
Let $ABC$ be a triangle with $\angle BAC = 90^o$ and $D$ be the point on the side $BC$ such that $AD \perp BC$. Let$ r, r_1$, and $r_2$ be the inradii of triangles $ABC, ABD$, and $ACD$, respectively. If $r, r_1$, and $r_2$ are positive integers and one of them is $5$, find the largest possible value of $r+r_1+ r_2$.
In the triangle $ABC$ angle $C$ is a right angle. On the side $AC$ point $D$ has been found, and on the segment $BD$ point K has been found such that $\angle ABC = \angle KAD = \angle AKD$. Prove that $BK = 2DC$.
Inside an isosceles right triangle $ABC$ with hypotenuse $AB$ a point $M$ is taken such that the angle $\angle MAB$ is $15 ^o$ larger than the angle $\angle MAC$ , and the angle $\angle MCB$ is $15^o$ larger than the angle $\angle MBC$. Find the angle $\angle BMC$ .
On the sides of triangle $ABC$, isosceles right-angled triangles $AUB, CVB$, and $AWC$ are placed. These three triangles have their right angles at vertices $U, V$ , and $W$, respectively. Triangle $AUB$ lies completely inside triangle $ABC$ and triangles $CVB$ and $AWC$ lie completely outside $ABC$. See the figure. Prove that quadrilateral $UVCW$ is a parallelogram.
[asy]
import markers;
unitsize(1.5 cm);
pair A, B, C, U, V, W;
A = (0,0);
B = (2,0);
C = (1.7,2.5);
U = (B + rotate(90,A)*(B))/2;
V = (B + rotate(90,C)*(B))/2;
W = (C + rotate(90,A)*(C))/2;
draw(A--B--C--cycle);
draw(A--W, StickIntervalMarker(1,1,size=2mm));
draw(C--W, StickIntervalMarker(1,1,size=2mm));
draw(B--V, StickIntervalMarker(1,2,size=2mm));
draw(C--V, StickIntervalMarker(1,2,size=2mm));
draw(A--U, StickIntervalMarker(1,3,size=2mm));
draw(B--U, StickIntervalMarker(1,3,size=2mm));
draw(rightanglemark(A,U,B,5));
draw(rightanglemark(B,V,C,5));
draw(rightanglemark(A,W,C,5));
dot("$A$", A, S);
dot("$B$", B, S);
dot("$C$", C, N);
dot("$U$", U, NE);
dot("$V$", V, NE);
dot("$W$", W, NW);
[/asy]
The triangle was divided into five triangles similar to it. Is it true that the original triangle is right-angled?
(S. Markelov)
A rectangle $ABEF$ is drawn on the leg $AB$ of a right triangle $ABC$, whose apex $F$ is on the leg $AC$. Let $X$ be the intersection of the diagonal of the rectangle $AE$ and the hypotenuse $BC$ of the triangle. In what ratio does point $X$ divide the hypotenuse $BC$ if it is known that $| AC | = 3 | AB |$ and $| AF | = 2 | AB |$?
Construct a right triangle given the hypotenuse and the median drawn to the leg.
The length of the hypotenuse $BC$ of a right triangle $ABC$ is $a$, and on it the points $M$ and $N$ are taken such that $BM = NC = k$, with $k < a/2$. Assuming that (only) the data $a$ and $k$ are known, calculate:
a) The value of the sum of the squares of the lengths $AM$ and $AN$.
b) The ratio of the areas of triangles $ABC$ and $AMN$.
c) The area enclosed by the circle that passes through the points $A, M' , N'$ , where $M'$ is the orthogonal projection of $M$ onto $AC$ and $N'$ that of $N$ onto $AB$.
Consider a right triangle $ABC$, where $A$ is the right angle, and $AC > AB$. Points $E$ on $AC$ and $D$ on $BC$ are chosen so that$ AB = AE = BD$. Prove that the triangle $ADE$ is right if and only if the ratio $AB : AC : BC$ of sides of the triangle $ABC$ is $3 : 4 : 5$.
(A. Parovan)
Let $D$ be the point different from $B$ on the hypotenuse $AB$ of a right triangle $ABC$ such that $|CB| = |CD|$. Let $O$ be the circumcenter of triangle $ACD$. Rays $OD$ and $CB$ intersect at point $P$, and the line through point $O$ perpendicular to side AB and ray $CD$ intersect at point $Q$. Points $A, C, P, Q$ are concyclic. Does this imply that $ACPQ$ is a square?
Consider a right -angle triangle $ABC$ with $A=90^{o}$, $AB=c$ and $AC=b$. Let $P\in AC$ and $Q\in AB$ such that $\angle APQ=\angle ABC$ and $\angle AQP = \angle ACB$. Calculate $PQ+PE+QF$, where $E$ and $F$ are the projections of $B$ and $Q$ onto $BC$, respectively.
In a right-angled triangle, $a$ and $b$ denote the lengths of the two catheti. A circle with radius $r$ has the center on the hypotenuse and touches both catheti. Show that $\frac{1}{a}+\frac{1}{b}=\frac{1}{r}$.
Let \( I \) and \( O \) be the incenter and circumcenter of the right triangle \( ABC \) (\( \angle C = 90^\circ \)), and let \( K \) be the tangency point of the incircle with \( AC \). Let \( P \) and \( Q \) be the points where the circumcircle of triangle \( AOK \) intersects \( OC \) and the circumcircle of triangle \( ABC \), respectively. Prove that points \( C, I, P, \) and \( Q \) are concyclic.
[i]Proposed by Mykhailo Sydorenko[/i]
Let $ABC$ be a triangle with $\angle A = 90^o$ and $\angle B < \angle C$. The tangent at $A$ to the circumcircle $k$ of $\vartriangle ABC$ intersects line $BC$ at $D$. Let $E$ be the reflection of $A$ in $BC$. Also, let $X$ be the feet of the perpendicular from $A$ to $BE$ and let $Y$ be the midpoint of $AX$. Line $BY$ meets $k$ again at $Z$. Prove that line $BD$ is tangent to the circumcircle of $\vartriangle ADZ$.
Let $a, b, c$ be side lengths of a right triangle. Determine the minimum possible value of $\frac{a^3 + b^3 + c^3}{abc}$.