Let $A B C$ be an acut-angles triangle of incenter $I$, circumcenter $O$ and inradius $r.$ Let $\omega$ be the inscribed circle of the triangle $A B C$. $A_1$ is the point of $\omega$ such that $A IA_1O$ is a convex trapezoid of bases $A O$ and $IA_1$. Let $\omega_1$ be the circle of radius $r$ which goes through $A_1$, tangent to the line $A B$ and is different from $\omega$ . Let $\omega_2$ be the circle of radius $r$ which goes through $A_1$, is tangent to the line $A C$ and is different from $\omega$ . Circumferences $\omega_1$ and $\omega_2$ they are cut at points $A_1$ and $A_2$. Similarly are defined points $B_2$ and $C_2$. Prove that the lines $A A_2, B B_2$ and $CC2$ they are concurrent.

The difference of two angles of a triangle is greater than $90^{\circ}$. Prove that the ratio of its circumradius and inradius is greater than $4$.

Let $ABC$ be a triangle with $\angle B=90$. The incircle of $ABC$ touches the side $BC$ at $D$. The incenters of triangles $ABD$ and $ADC$ are $X$ and $Z$ , respectively. The lines $XZ$ and $AD$ are intersecting at the point $K$. $XZ$ and circumcircle of $ABC$ are intersecting at $U$ and $V$. Let $M$ be the midpoint of line segment $[UV]$ . $AD$ intersects the circumcircle of $ABC$ at $Y$ other than $A$. Prove that $|CY|=2|MK|$ .

Inside right angle $XAY$, where $A$ is the vertex, is a semicircle $\Gamma$ whose center lies on $AX$ and that is tangent to $AY$ at the point $A$. Describe a ruler-and-compass construction for the tangent to $\Gamma$ such that the triangle enclosed by the tangent and angle $XAY$ has minimum area.

A triangle has semi-perimeter $s$, circumradius $R$ and inradius $r$. Show that it is right-angled iff $2R = s - r$.

We are given trapezoid $ABCD$ and point $M$ on the intersection of its diagonals. The parallel sides are $AD$ and $BC$ and it is known that $AB$ is perpendicular to $AD$ and that the trapezoid can have an inscribed circle. If the radius of this inscribed circle is $R$ find the area of triangle $DCM$ .

Points $B = B_1 , B_2, \cdots , B_n , B_{n+1} = C$ are chosen on side $BC$ of a triangle $ABC$ in that order. Let $r_j$ be the inradius of triangle $AB_jB_{j+1}$ for $j = 1, \cdots, n$ , and $r$ be the inradius of $\triangle ABC$. Show that there is a constant $\lambda$ independent of $n$ such that : \[(\lambda -r_1)(\lambda -r_2)\cdots (\lambda -r_n) =\lambda^{n-1}(\lambda -r)\]

The sides of a triangle are three consecutive integers and its inradius is $4$. Find the circumradius.

Let $ E$ be the intersection of the diagonals of the convex quadrilateral $ ABCD$. The perimeters of $ \triangle AEB$, $ \triangle BEC$, $ \triangle CED$, and $ \triangle DEA$ are all same. If inradii of $ \triangle AEB$, $ \triangle BEC$, $ \triangle CED$ are $ 3,4,6$, respectively, then inradius of $ \triangle DEA$ will be ? $\textbf{(A)}\ \frac {9}{2} \qquad\textbf{(B)}\ \frac {7}{2} \qquad\textbf{(C)}\ \frac {13}{3} \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{None}$

Let $\Delta ABC$ be a triangle with angle $\angle CAB=60^{\circ}$, let $D$ be the intersection point of the angle bisector at $A$ and the side $BC$, and let $r_B,r_C,r$ be the respective radii of the incircles of $ABD$, $ADC$, $ABC$. Let $b$ and $c$ be the lengths of sides $AC$ and $AB$ of the triangle. Prove that \[ \frac{1}{r_B} +\frac{1}{r_C} ~=~ 2\cdot\left( \frac1r +\frac1b +\frac1c\right)\]

In the triangle$ABC$ ($AC > AB$), point $N$ is the midpoint of $BC$, and $I$ is the intersection point of the angle bisectors. Ray $AI$ intersects the circumscribed circle of triangle $ABC$ at point $W$, a perpendicular $WF$ is drawn from it on side $AC$. Find the length of the segment $CF$ , if the radius of the circle inscribed in the triangle $ABC$ is equal to $r$ and $\angle INB = 45^o$. (Gryhoriy Filippovskyi)

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.

Let $ABC$ be a triangle, let $O$ be its circumcenter, and let $H$ be its orthocenter. Let $P$ be a point on the segment $OH$. Prove that $6r\leq PA+PB+PC\leq 3R$, where $r$ is the inradius and $R$ the circumradius of triangle $ABC$. [b]Moderator edit:[/b] This is true only if the point $P$ lies inside the triangle $ABC$. (Of course, this is always fulfilled if triangle $ABC$ is acute-angled, since in this case the segment $OH$ completely lies inside the triangle $ABC$; but if triangle $ABC$ is obtuse-angled, then the condition about $P$ lying inside the triangle $ABC$ is really necessary.)

The diagonals of the square $ABCD$ intersect at $P$ and the midpoint of the side $AB$ is $E$. Segment $ED$ intersects the diagonal $AC$ at point $F$ and segment $EC$ intersects the diagonal $BD$ at $G$. Inside the quadrilateral $EFPG$, draw a circle of radius $r$ tangent to all the sides of this quadrilateral. Prove that $r = | EF | - | FP |$.

Find the radius of the inscribed circle of a triangle with sides of length $40$, $42$, and $58$.

How many integer triangles are there with inradius $1$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{Infinite} $

(B.Frenkin) The product of two sides in a triangle is equal to $ 8Rr$, where $ R$ and $ r$ are the circumradius and the inradius of the triangle. Prove that the angle between these sides is less than $ 60^{\circ}$.

In a triangle with sides $a, b, c$ the side $a$ is the arithmetic mean of $b$ and $c$. Prove that: a) $0^o \le A \le 60^o$. b) The height relative to side $a$ is three times the inradius $r$. c) The distance from the circumcenter to side $a$ is $R - r$, where $R$ is the circumradius.

An arbitrary triangle $ABC$ is given. Construct a straight line passing through vertex $B$ and dividing it into two triangles, the radii of the inscribed circles of which are equal. (M. Volchkevich)

For a given triangle $ ABC$, let $ X$ be a variable point on the line $ BC$ such that $ C$ lies between $ B$ and $ X$ and the incircles of the triangles $ ABX$ and $ ACX$ intersect at two distinct points $ P$ and $ Q.$ Prove that the line $ PQ$ passes through a point independent of $ X$.

Let $n$ be positive integer and $S$= {$0,1,…,n$}, Define set of point in the plane. $$A = \{(x,y) \in S \times S \mid -1 \leq x-y \leq 1 \} $$, We want to place a electricity post on a point in $A$ such that each electricity post can shine in radius 1.01 unit. Define minimum number of electricity post such that every point in $A$ is in shine area

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.$

Let $ABCD$ be a convex quadrilateral. Prove that the incircles of the triangles $ABC$, $BCD$, $CDA$ and $DAB$ have a point in common if, and only if, $ABCD$ is a rhombus.

Let $r$ be the inradius of triangle $ABC$. Take a point $D$ on side $BC$, and let $r_1$ and $r_2$ be the inradii of triangles $ABD$ and $ACD$. Prove that $r$, $r_1$, and $r_2$ can always be the side lengths of a triangle.

Prove that the sum of the squares of the medians of a triangle is at least $ 9/4 $ if the circumradius of the triangle, the area of the triangle and the inradius of the triangle (in this order) are in arithmetic progression. [i]Dumitru Crăciun[/i]