Found problems: 325
let $ABC$ be a right angled triangle with inradius $1$
find the minimum area of triangle $ABC$
The circle inscribed in triangle $ABC$ touches $AC$ at point $F$. The perpendicular from point $F$ on $BC$ intersects the bisector of angle $C$ at point $N$. Prove that segment $FN$ is equal to the radius of the circle inscribed in triangle $ABC$.
(Oleksii Karliuchenko)
Given a circumscribed quadrilateral $ABCD$.
Prove that its inradius is smaller than the sum of the inradii of triangles $ABC$ and $ACD$.
$P$ is a point inside the triangle $ABC$ is a triangle. The distance of $P$ from the lines $BC, CA, AB$ is $d_a, d_b, d_c$ respectively. If $r$ is the inradius, show that $$\frac{2}{ \frac{1}{d_a} + \frac{1}{d_b} + \frac{1}{d_c}} < r < \frac{d_a + d_b + d_c}{2}$$
Prove that a triangle $ABC$ is right-angled if and only if
\[\sin A + \sin B + \sin C = \cos A + \cos B + \cos C + 1\]
Let $r$ be a positive real number. Prove that the number of right triangles with prime positive integer sides that have an inradius equal to $r$ are zero or a power of $2$.
[hide=original wording]Seja r um numero real positivo. Prove que o numero de triangulos retangulos com lados inteiros positivos primos entre si que possuem inraio igual a r e zero ou uma potencia de 2.[/hide]
Consider a circle $\Gamma$, a point $A$ on its exterior, and the points of tangency $B$ and $C$ from $A$ to $\Gamma$. Let $P$ be a point on the segment $AB$, distinct from $A$ and $B$, and let $Q$ be the point on $AC$ such that $PQ$ is tangent to $\Gamma$. Points $R$ and $S$ are on lines $AB$ and $AC$, respectively, such that $PQ\parallel RS$ and $RS$ is tangent to $\Gamma$ as well. Prove that $[APQ]\cdot[ARS]$ does not depend on the placement of point $P$.
Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. let $V_1$ be the volume of the cone and $V_2$ be the volume of the cylinder.
a) Prove that $V_1 \neq V_2$;
b) Find the smallest number $k$ for which $V_1=kV_2$; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone.
Find the maximum possible value of the inradius of a triangle whose vertices lie in the interior, or on the boundary, of a unit square.
Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that
\[r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}\]
where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.
In right $ \triangle ABC$ with hypotenuse $ \overline{AB}$, $ AC \equal{} 12$, $ BC \equal{} 35$, and $ \overline{CD}$ is the altitude to $ \overline{AB}$. Let $ \omega$ be the circle having $ \overline{CD}$ as a diameter. Let $ I$ be a point outside $ \triangle ABC$ such that $ \overline{AI}$ and $ \overline{BI}$ are both tangent to circle $ \omega$. The ratio of the perimeter of $ \triangle ABI$ to the length $ AB$ can be expressed in the form $ \displaystyle\frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.
Let ABC be a triangle and let $r, r_a, r_b, r_c$ denote the inradius and ex-radii opposite to the vertices $A, B, C$, respectively. Suppose that $a>r_a, b>r_b, c>r_c$. Prove that
[b](a)[/b] $\triangle ABC$ is acute.
[b](b)[/b] $a+b+c > r+r_a+r_b+r_c$.
Incircle of triangle $ ABC$ touches $ AB,AC$ at $ P,Q$. $ BI, CI$ intersect with $ PQ$ at $ K,L$. Prove that circumcircle of $ ILK$ is tangent to incircle of $ ABC$ if and only if $ AB\plus{}AC\equal{}3BC$.
[b]Q14.[/b] Let be given a trinagle $ABC$ with $\angle A=90^o$ and the bisectrices of angles $B$ and $C$ meet at $I$. Suppose that $IH$ is perpendicular to $BC$ ($H$ belongs to $BC$). If $HB=5 \text{cm}, \; HC=8 \text{cm}$, compute the area of $\triangle ABC$.
Let $ ABCD$ be a quadrilateral in which $ AB$ is parallel to $ CD$ and perpendicular to $ AD; AB \equal{} 3CD;$ and the area of the quadrilateral is $ 4$. if a circle can be drawn touching all the four sides of the quadrilateral, find its radius.
$L$ and $M$ are two parallel lines a distance $d$ apart. Given $r$ and $x$, construct a triangle $ABC$, with $A$ on $L$, and $B$ and $C$ on $M$, such that the inradius is $r$, and angle $A = x$. Calculate angles $B$ and $C$ in terms of $d$, $r$ and $x$. If the incircle touches the side $BC$ at $D$, find a relation between $BD$ and $DC$
The segment $AB$ divides the square into two parts, in each of which a circle can be inscribed.
The radii of these circles are equal to $r_1$ and $r_2$ respectively, where $r_1> r_2$.
Find the length of $AB$.
From point $M$ in triangle $ABC$ perpendiculars are dropped to each altitude. It can be shown that each of the line segments of altitudes, measured between the vertex and the foot of the perpendicular drawn to it, are of equal length. Prove that these lengths are each equal to the diameter of the circle inscribed in the triangle.
Among all triangles having (i) a fixed angle $A$ and (ii) an inscribed circle of fixed radius $r$, determine which triangle has the least minimum perimeter.
Let $ABCD$ be a rectangle, and let $\omega$ be a circular arc passing through the points $A$ and $C$.
Let $\omega_{1}$ be the circle tangent to the lines $CD$ and $DA$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$.
Similiarly let $\omega_{2}$ be the circle tangent to the lines $AB$ and $BC$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$.
Denote by $r_{1}$ and $r_{2}$ the radii of the circles $\omega_{1}$ and $\omega_{2}$, respectively, and by $r$ the inradius of triangle $ABC$.
[b](a)[/b] Prove that $r_{1}+r_{2}=2r$.
[b](b)[/b] Prove that one of the two common internal tangents of the two circles $\omega_{1}$ and $\omega_{2}$ is parallel to the line $AC$ and has the length $\left|AB-AC\right|$.
Let K be a point on the side BC of the triangle ABC. The incircles of the triangles ABK and ACK touch BC at points M and N, respectively. Show that [tex]BM\cdot CN>KM \cdot KN[/tex].
Let $ABC$ be an equilateral triangle of sidelength 2 and let $\omega$ be its incircle.
a) Show that for every point $P$ on $\omega$ the sum of the squares of its distances to $A$, $B$, $C$ is 5.
b) Show that for every point $P$ on $\omega$ it is possible to construct a triangle of sidelengths $AP$, $BP$, $CP$. Also, the area of such triangle is $\frac{\sqrt{3}}{4}$.
Let the diagonals of the square $ABCD$ intersect at $S$ and let $P$ be the midpoint of $AB$. Let $M$ be the intersection of $AC$ and $PD$ and $N$ the intersection of $BD$ and $PC$. A circle is incribed in the quadrilateral $PMSN$. Prove that the radius of the circle is $MP- MS$.
A point $ M$ is chosen on the side $ AC$ of the triangle $ ABC$ in such a way that the radii of the circles inscribed in the triangles $ ABM$ and $ BMC$ are equal. Prove that
\[ BM^{2} \equal{} X \cot \left( \frac {B}{2}\right)
\]
where X is the area of triangle $ ABC.$
$ABC$ 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.