Found problems: 333
The sides of a right triangle are $a, a+d,$ and $a+2d$, with $a$ and $d$ both positive. The ratio of $a$ to $d$ is:
$ \textbf{(A)}\ 1:3 \qquad\textbf{(B)}\ 1:4 \qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 3:1\qquad\textbf{(E)}\ 3:4 $
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 $\angle CAP$
Let $ABC $be a right triangle with $\angle C = 90^o$ and let $D$ be the foot of the altitude from $C$. Let $E$ be the centroid of triangle $ACD$ and let $F$ be the centroid of triangle $BCD$. The point $P$ satisfies $\angle CEP = 90^o$ and $|CP| = |AP|$, while point $Q$ satisfies $\angle CFQ = 90^o$ and $|CQ| = |BQ|$.
Prove that $PQ$ passes through the centroid of triangle $ABC$.
Given a right triangle $ABC$ with the hypotenuse $AB$. Let $K$ be any interior point of triangle $ABC$ and points $L, M$ are symmetric of point $K$ wrt lines $BC, AC$ respectively. Specify all possible values for $S_{ABLM} / S_{ABC}$, where $S_{XY ... Z}$ indicates the area of the polygon $XY...Z$ .
Let $ABC$ be a triangle with $\angle CAB > 45^o$ and $\angle CBA > 45^o$. Construct an isosceles right angled triangle $RAB$ with $AB$ as its hypotenuse and $R$ inside $ABC$. Also construct isosceles right angled triangles $ACQ$ and $BCP$ having $AC$ and $BC$ respectively as their hypotenuses and lying entirely outside $ABC$. Show that $CQRP$ is a parallelogram.
Given a right triangle $ABC$, the point $M$ is the midpoint of the hypotenuse $AB$. A circle is circumscribed around the triangle $BCM$, which intersects the segment $AC$ at a point $Q$ other than $C$. It turned out that the segment $QA$ is twice as large as the side $BC$. Find the acute angles of triangle $ABC$.
(Mykola Moroz)
$ABC$ is a triangle with $\angle B = 90^o$ and altitude $BH$. The inradii of $ABC, ABH, CBH$ are $r, r_1, r_2$. Find a relation between them.
On side $AB$ of triangle $ABC$ a point $P$ is taken such that $AP = 2PB$. It is known that $CP = 2PQ$ where $Q$ is the midpoint of $AC$. Prove that $ABC$ is a right triangle.
On the sides of the triangle $ABC$ externally constructed right triangles $ABC_1$, $BCA_1$, $CAB_1$. Prove that the points of intersection of the medians of the triangles $ABC$ and $A_1B_1C_1$ coincide.
Given a right triangle $ABC$ ($\angle C=90^o$). The $C$-excircle touches the hypotenuse $AB$ at point $C_1, A_1$ is the touchpoint of $B$-excircle with line $BC, B_1$ is the touchpoint of $A$-excircle with line $AC$. Find the angle $\angle A_1C_1B_1$.
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
\[ \frac {E}{E_1} \geq 2.
\]
In a right-angled triangle, the sum $a + b$ of the sides enclosing the right angle equals $24$ while the length of the altitude $h_c$ on the hypotenuse $c$ is $7$. Determine the length of the hypotenuse.
Let $n = p_1p_2... p_{2017}$ be the positive integer where $p_1, p_2, ..., p_{2017}$ are $2017$ distinct odd primes. A triangle is called [i]nice [/i] if it is a right triangle with integer side lengths and the inradius is $n$. Find the number of nice triangles (two triangles are consider different if their tuples of length of sides are different)
Consider the isosceles right triangle$ ABC, \angle A = 90^o$, and point $D \in (AB)$ such that $AD = \frac13 AB$. In the half-plane determined by the line $AB$ and point $C$ , consider a point $E$ such that $\angle BDE = 60^o$ and $\angle DBE = 75^o$. Lines $BC$ and $DE$ intersect at point $G$, and the line passing through point $G$ parallel to the line $AC$ intersects the line $BE$ at point $H$. Prove that the triangle $CEH$ is equilateral.
Let $\triangle ABC$ be a right triangle at the vertex $A$ such that the side $AB$ is shorter than the side $AC$.
Let $D$ be the foot of the altitude from $A$ to $BC$ and $M$ the midpoint of $BC$. Let $E$ be a point on the ray $AB$, outside of the segment $AB$. Line $ED$ intersects the segment $AM$ at the point $F$. Point $H$ is on the side $AC$ such that $\angle EFH=90^{\circ}$. Suppose that $ED=FH$. Find the measure of the angle $\angle AED$.
In an isosceles right-angled triangle $ABC$, the right angle is at $A$. $D$ lies so on the side $BC$ that $2CD = DB$. Let $E$ be the projection of $B$ onto $AD$. What is the measure fof angle $\angle CED $?
Consider the right triangle $\vartriangle ABC$ right at $A$ and let $D$ be a point on the hypotenuse $BC$. Consider the line that passes through the incenters of $\vartriangle ABD$ and $\vartriangle ACD$, and let $K$ and $ L$ the intersections of said line with $AB$ and $AC$ respectively. Show that if $AK = AL$ then $D$ is the foot of the altitude on the hypotenuse.
Prove that if $a$ and $b$ are legs, $c$ is the hypotenuse of a right triangle, then the radius of a circle inscribed in this triangle can be found by the formula $r = \frac12 (a + b - c)$.
Can a tetrahedron scan turn out to be a triangle with sides $3, 4$ and $5$ (a tetrahedron can be cut only along the edges)?
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?
The point ${D}$ inside the equilateral triangle ${\triangle ABC}$ satisfies ${\angle ADC = 150^o}$. Prove that a triangle with side lengths ${|AD|, |BD|, |CD|}$ is necessarily a right-angled triangle.
A circle with center $O$ passes through the ends of the hypotenuse of a right-angled triangle and intersects its legs at points $M$ and $K$. Prove that the distance from point $O$ to line $MK$ is half the hypotenuse.
Four right triangles, each with the sides $1$ and $2$, are assembled to a figure as shown.
How large a fraction does the area of the small circle make up of that of the big one?
[img]https://1.bp.blogspot.com/-XODK1XKCS0Q/XzXDtcA-xAI/AAAAAAAAMWA/zSLPpf3IcX0rgaRtOxm_F2begnVdUargACLcBGAsYHQ/s0/2010%2BMohr%2Bp1.png[/img]
Construct a right triangle given the lengths of segments of the medians $m_a,m_b$ corresponding on its legs.
Outside the square $ABCD$ is constructed the right isosceles triangle $ABD$ with hypotenuse $[AB]$. Let $N$ be the midpoint of the side $[AD]$ and ${M} = CE \cap AB$, ${P} = CN \cap AB$ , ${F} = PE \cap MN$. On the line $FP$ the point $Q$ is considered such that the $[CE$ is the bisector of the angle $QCB$. Prove that $MQ \perp CF$.