Suppose three circles of radius $5$ intersect at a common point. If the three (other) pairwise intersections between the circles form a triangle of area $ 8$, find the radius of the smallest possible circle containing all three circles.

Three equal circles of radius $r$ each pass through the centres of the other two. What is the area of intersection that is common to all the three circles?

Points $A_1,B_1,C_1$ are selected on the sides $BC$,$CA$,$AB$ respectively of an equilateral triangle $ABC$ in such a way that the inradii of the triangles $C_1AB_1$, $A_1BC_1$, $B_1CA_1$ and $A_1B_1C_1$ are equal. Prove that $A_1,B_1,C_1$ are the midpoints of the corresponding sides.

Two equal circles $\Omega_1$ and $\Omega_2$ intersect at points $A$ and $B$, and $M$ is the midpoint of $AB$. Two rays were drawn from $M$, lying in the same half-plane wrt $AB$ (see figure). The first ray intersects the circles $\Omega_1$ and $\Omega_2$ at points $X_1$ and $X_2$, and the second ray intersects them at points $Y_1$ and $Y_2$, respectively. Let $C$ be the intersection point of straight lines $AX_1$ and $BY_2$, and let $D$ be the intersection point of straight lines $AX_2$ and $BY_1$. Prove that $CD \parallel AB$. [img]https://cdn.artofproblemsolving.com/attachments/4/a/fae047c3956d8b30f15a9d88e8d12e5f4d48ec.png[/img]

Three equal circles $k_1, k_2, k_3$ intersect non-tangentially at a point $P$. Let $A$ and $B$ be the centers of circles $k_1$ and $k_2$. Let $D$ and $C$ be the intersection of $k_3$ with $k_1$ and $k_2$ respectively, which is different from $P$. Show that $ABCD$ is a parallelogram.

Two circles $c$ and $d$ are situated in the plane each outside the other. The points $C$ and $D$ are located on circles $c$ and $d$ respectively, so as to be as far apart as possible. Two smaller circles are constructed inside $c$ and $d$. Of these the first circle touches $c$ and the two tangents drawn from $C$ to $d$, while the second circle touches $d$ and the two tangents from $D$ to $c$. Prove that the small circles are equal. (J. Tabov, Sofia)

Two circles of the same radii intersect in two distinct points $P$ and $Q$. A line passing through $P$, not touching any of the circles, intersects the circles again at $A$ and $B$. Prove that $Q$ lies on the perpendicular bisector of $AB$.

In the figure shown, the small circles have radius $1$. Calculate the area of the gray part of the figure. [img]https://1.bp.blogspot.com/-oy-WirJ6u9o/XzcFc3roVDI/AAAAAAAAMX8/qxNy5I_0RWUOxl-ZE52fnrwo0v0T7If9QCLcBGAsYHQ/s0/1998%2BMohr%2Bp1.png[/img]

$ABC$ is an acute-angled triangle with circumcenter $O$. The circumcircle of $ABO$ intersects$ AC$ and $BC$ at $M$ and $N$. Show that the circumradii of $ABO$ and $MNC$ are the same.

Eight circles of radius $r$ located in a right triangle $ABC$ (angle $C$ is right) as shown in figure (each of the circles touches the respactive sides of the triangle and the other circles). Find the radius of the inscribed circle of triangle $ABC$. [img]https://cdn.artofproblemsolving.com/attachments/4/7/1b1cd7d6bc7f5004b8e94468d723ed16e9a039.png[/img]

Points $A, B$ move with equal speeds along two equal circles. Prove that the perpendicular bisector of $AB$ passes through a fixed point.

There is a square $ABCD$ in the plane with $|AB|=a$. Determine the smallest possible radius value of three equal circles to cover a given square.

Three circles of radius $1$ are inscribed in a square of side length $s$, such that the circles do not overlap or coincide with each other. What is the minimum $s$ where such a configuration is possible?

a) Does there exist a convex quadrangle $ABCD$ satisfying the following conditions (1) $ABCD$ is not cyclic; (2) the sides $AB, BC, CD$ and $DA$ have pairwise different lengths; (3) the circumradii of the triangles $ABC, ADC, BAD$ and $BCD$ are equal? b) Does there exist such a non-convex quadrangle?

Given a triangle$ABC$ with $AB<BC<AC$ inscribed in circle $(c)$. The circle $c(A,AB)$ (with center $A$ and radius $AB$) interects the line $BC$ at point $D$ and the circle $(c)$ at point $H$. The circle $c(A,AC)$ (with center $A$ and radius $AC$) interects the line $BC$ at point $Z$ and the circle $(c)$ at point $E$. Lines $ZH$ and $ED$ intersect at point $T$. Prove that the circumscribed circles of triangles $TDZ$ and $TEH$ are equal.

Given three circumferences of the same radius in a plane. a) All three are crossing in one point $K$. Consider three arcs $AK,CK,EK$ : the $A,C,E$ are the points of the circumferences intersection and the arcs are taken in the clockwise direction. Every arc is inside one circle, outside the second and on the border of the third one. Prove that the sum of the arcs is $180$ degrees. b) Consider the case, when the three circles give a curvilinear triangle $BDF$ as their intersection (instead of one point $K$). The arcs are taken in the clockwise direction. Every arc is inside one circle, outside the second and on the border of the third one. Prove that the sum of the $AB, CD$ and $EF$ arcs is $180$ degrees.

The figure shows a square and three circles of equal radius tangent to each other and square passes. Find the radius of the circles if the square length is $1$. [img]http://3.bp.blogspot.com/-iIjwupkz7DQ/XnrIRhKIJnI/AAAAAAAALhA/clERrIDqEtcujzvZk_qu975wsTjKaxCLQCK4BGAYYCw/s400/97%2Bestonia%2Bopen%2Bs2.3.png[/img]

An arbitrary triangle $ABC$ is given. Construct a line that divides it into two polygons, which have equal radii of the circumscribed circles. (L. Blinkov)

On a line $\ell$ there exists $3$ points $A, B$, and $C$ where $B$ is located between $A$ and $C$. Let $\Gamma_1, \Gamma_2, \Gamma_3$ be circles with $AC, AB$, and $BC$ as diameter respectively; $BD$ is a segment, perpendicular to $\ell$ with $D$ on $\Gamma_1$. Circles $\Gamma_4, \Gamma_5, \Gamma_6$ and $\Gamma_7$ satisfies the following conditions: $\bullet$ $\Gamma_4$ touches $\Gamma_1, \Gamma_2$, and$ BD$. $\bullet$ $\Gamma_5$ touches $\Gamma_1, \Gamma_3$, and $BD$. $\bullet$ $\Gamma_6$ touches $\Gamma_1$ internally, and touches $\Gamma_2$ and $\Gamma_3$ externally. $\bullet$ $\Gamma_7$ passes through $B$ and the tangent points of $\Gamma_2$ with $\Gamma_6$, and $\Gamma_3$ with $\Gamma_6$. Show that the circles $\Gamma_4, \Gamma_5$, and $\Gamma_7$ are congruent.

Let $ABCD$ be a parallelogram with at least an angle not equal to $90^o$ and $k$ the circumcircle of the triangle $ABC$. Let $E$ be the diametrically opposite point of $B$. Show that the circumcircle of the triangle $ADE$ and $k$ have the same radius.

Given a triangle$ABC$ with $AB<BC<AC$ inscribed in circle $(c)$. The circle $c(A,AB)$ (with center $A$ and radius $AB$) interects the line $BC$ at point $D$ and the circle $(c)$ at point $H$. The circle $c(A,AC)$ (with center $A$ and radius $AC$) interects the line $BC$ at point $Z$ and the circle $(c)$ at point $E$. Lines $ZH$ and $ED$ intersect at point $T$. Prove that the circumscribed circles of triangles $TDZ$ and $TEH$ are equal.

On the side $CD$ of the square $ABCD$, the point $F$ is chosen and the equal squares $DGFE$ and $AKEH$ are constructed ($E$ and $H$ lie inside the square). Let $M$ be the midpoint of $DF$, $J$ is the incenter of the triangle $CFH$. Prove that: a) the points $D, K, H, J, F$ lie on the same circle; b) the circles inscribed in triangles $CFH$ and $GMF$ have the same radii.

Three circles through the point $O$ and of radius $r$ intersect pairwise in the additional points $A$,$B$,$C$. Prove that the circle through the points $A$, $B$, and $C$ also has radius $r$.

In the figure shown, the small circles have radius $1$. Calculate the area of the gray part of the figure. [img]https://1.bp.blogspot.com/-oy-WirJ6u9o/XzcFc3roVDI/AAAAAAAAMX8/qxNy5I_0RWUOxl-ZE52fnrwo0v0T7If9QCLcBGAsYHQ/s0/1998%2BMohr%2Bp1.png[/img]

Let be given an isosceles triangle $ABC$ with the base $AB$. A point $P$ is chosen on the altitude $CD$ so that the incircles of $ABP$ and $PECF$ are congruent, where $E$ and $F$ are the intersections of $AP$ and $BP$ with the opposite sides of the triangle, respectively. Prove that the incircles of triangles $ADP$ and $BCP$ are also congruent.