Transform by inversion two concentric and coplanar circles into two equal.

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 $AB$.

Points $A_1,A_2,A_3,A_4$ are not concyclic, the same for points $B_1,B_2,B_3,B_4$. For all $i, j, k$ the circumradii of triangles $A_iA_jA_k$ and $B_iB_jB_k$ are equal. Can we assert that $A_iA_j=B_iB_j$ for all $i, j$'?

A circle $(K)$ is through the vertices $B, C$ of the triangle $ABC$ and intersects its sides $CA, AB$ respectively at $E, F$ distinct from $C, B$. Line segment $BE$ meets $CF$ at $G$. Let $M, N$ be the symmetric points of $A$ about $F, E$ respectively. Let $P, Q$ be the reflections of $C, B$ about $AG$. Prove that the circumcircles of triangles $BPM , CQN$ have radii of the same length. Trần Quang Hùng

Point $O$ is the intersection point of the heights of an acute triangle $\vartriangle ABC$. Prove that the three circles which pass: a) through $O, A, B$, b) through $O, B, C$, and c) through $O, C, A$, 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.

Let $I$ be the incenter of an acute riangle $ABC$. Let $C_A(A, AI)$ be the circle with center $A$ and radius $AI$. Circles $C_B(B, BI)$, $C_C(C, CI) $ are defined in an analogous way. Let $X, Y, Z$ be the intersection points of $C_B$ with $C_C$, $C_C$ with $C_A$, $C_A$ with $C_B$ respectively (different than $I$) . Show that if the radius of the circle that passes through the points $X, Y, Z$ is equal to the radius of the circle that passes through points $A$, $B$ and $C$ then triangle $ABC$ is equilateral.

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)

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]

Show that if a triangle has two excircles of the same size, then the triangle is isosceles. (Note: The excircle $ABC$ to the side $ a$ touches the extensions of the sides $AB$ and $AC$ and the side $BC$.)

Let $ \Gamma_1$ and $\Gamma_2$ be two different circles with the radius of same length and centers at points $O_1$ and $O_2$, respectively. Circles $\Gamma_1$ and $\Gamma_2$ are tangent at point $P$. The line $\ell$ passing through $O_1$ is tangent to $\Gamma_2$ at point $A$. The line $\ell$ intersects $\Gamma_1$ at point $X$ with $X$ between $A$ and $O_1$. Let $M$ be the midpoint of $AX$ and $Y$ the intersection of $PM$ and $\Gamma_2$ with $Y\ne P$. Prove that $XY$ is parallel to $O_1O_2$.

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.

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

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.

Find the total area of the shaded area in the figure if all circles have an equal radius $R$ and the centers of the outer circles divide into six equal parts of the middle circle. [img]http://3.bp.blogspot.com/-Ax0QJ38poYU/XovXkdaM-3I/AAAAAAAALvM/DAZGVV7TQjEnSf2y1mbnse8lL6YIg-BQgCK4BGAYYCw/s400/estonia%2B2000%2Bo.j.1.5.png[/img]

Prove that if $ H $ is the point of intersection of the altitudes of a non-right triangle $ ABC $, then the circumcircles of the triangles $ AHB $, $ BHC $, $ CHA $ and $ ABC $ are equal.

Three circles $\omega_1$, $\omega_2$, $\omega_3$ of radius $r$ pass through the point$ S$ and internally touch the circle $\omega$ of radius $R$ ($R > r$) at points $T_1$, $T_2$, $T_3$ respectively. Prove that the line $T_1T_2$ passes through the second (different from $S$) intersection point of the circles $\omega_1$ and $\omega_2$.

Let $AD$ be the common chord of two equal-sized circles $O_1$ and $O_2$. Let $B$ and $C$ be points on $O_1$ and $O_2$, respectively, so that $D$ lies on the segment $BC$. Assume that $AB = 15, AD = 13$ and $BC = 18$, what is the ratio between the inradii of $\vartriangle ABD$ and $\vartriangle ACD$?

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.

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]

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]

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.

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.

In the plane, the point $M$ is the midpoint of a line segment $[AB]$ and $\ell$ is an arbitrary line that has no has a common point with the line segment $[AB]$ (and is also not perpendicular to $[AB]$). The points $X$ and $Y$ are the perpendicular projections of $A$ and $B$ onto $\ell$, respectively. Show that the circumscribed circles of triangle $\vartriangle AMX$ and triangle $\vartriangle BMY$ have the same radius.