Two given circles $\alpha$ and $\beta$ intersect each other at two points. Find the locus of the centers of all circles that are orthogonal to both $\alpha$ and $\beta$.

Two circles intersect at points $P$ and $Q$. The straight line intersects these circles at points $A$, $B$, $C$, $D$, as shown in fig. . Prove that $\angle APB = \angle CQD$. [img]https://cdn.artofproblemsolving.com/attachments/1/a/a581e11be68bbb628db5b5b8e75c7ff6e196c5.png[/img]

Six small circles of radius $1$ are drawn so that they are all tangent to a larger circle, and two of them are tangent to sides $BC$ and $AD$ of a rectangle $ABCD$ at their midpoints $K$ and $L$. Each of the remaining four small circles is tangent to two sides of the rectangle. The large circle is tangent to sides $AB$ and $CD$ of the rectangle and cuts the other two sides. Find the radius of the large circle. [img]https://cdn.artofproblemsolving.com/attachments/b/4/a134da78d709fe7162c48d6b5c40bd1016c355.png[/img]

7. The four Vertices of a quadrilateral [i]ABCD[/i] lie on the circle with diameter [i]AB[/i]. The diagonals of [i]ABCD[/i] intersect at [i]E[/i], and the lines [i]AD[/i] and [i]BC[/i] intersect at [i]F[/i]. Line [i]FE[/i] meets [i]AB[/i] at [i]K[/i] and line [i]DK[/i] meets the circle again at [i]L[/i]. Prove that [i]CL[/i] is perpendicular to [i]AB[/i].

Two different circles $\omega_1$ ,$\omega_2$, with centers $O_1, O_2$ respectively intersect at the points $A, B$. The line $O_1B$ intersects $\omega_2$ at the point $F (F \ne B)$, and the line $O_2B$ intersects $\omega_1$ at the point $E (E\ne B)$. A line was drawn through the point $B$, parallel to the $EF$, which intersects $\omega_1$ at the point $M (M \ne B)$, and $\omega_2$ at the point $N (N\ne B)$. Prove that the lines $ME, AB$ and $NF$ intersect at one point.

Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$. (1) Prove that there exists an equilateral triangle whose vertices lie in different discs. (2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$. [i]Radu Gologan, Romania[/i] [hide="Remark"] The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url]. [/hide]

Let $ABC$ be an equilateral triangle and let $D$ be an inner point of the side $BC$. A circle is tangent to $BC$ at $D$ and intersects the sides $AB$ and $AC$ in the inner points $M, N$ and $P, Q$ respectively. Prove that $|BD| + |AM| + |AN| = |CD| + |AP| + |AQ|$.

A chord $AB$ is drawn in the circle, on which the point $P$ is selected in such a way that $AP = 2PB$. The chord $DE$ is perpendicular to the chord $AB $ and passes through the point $P$. Prove that the midpoint of the segment $AP$ is the orthocener of the triangle $AED$.

A circle $S$ bisects a circle $S'$ if it cuts $S'$ at opposite ends of a diameter. $S_A$, $S_B$,$S_C$ are circles with distinct centers $A, B, C$ (respectively). Show that $A, B, C$ are collinear iff there is no unique circle $S$ which bisects each of $S_A$, $S_B$,$S_C$ . Show that if there is more than one circle $S$ which bisects each of $S_A$, $S_B$,$S_C$ , then all such circles pass through two fixed points. Find these points. [b]Original Statement:[/b] A circle $S$ is said to cut a circle $\Sigma$ [b]diametrically[/b] if and only if their common chord is a diameter of $\Sigma.$ Let $S_A, S_B, S_C$ be three circles with distinct centres $A,B,C$ respectively. Prove that $A,B,C$ are collinear if and only if there is no unique circle $S$ which cuts each of $S_A, S_B, S_C$ diametrically. Prove further that if there exists more than one circle $S$ which cuts each $S_A, S_B, S_C$ diametrically, then all such circles $S$ pass through two fixed points. Locate these points in relation to the circles $S_A, S_B, S_C.$

Circle of radius $R_1$ is inscribed in an acute angle $\alpha$. Second circle with radius $R_2$ touches one of the sides forming the angle $\alpha$ in same point as first circle and intersects the second side in points $A$ and $B$, such that centers of both circles lie inside angle $\alpha$. Prove that $$AB=4\cos{\frac{\alpha}{2}}\sqrt{(R_2-R_1)\left(R_1 \cos^2 \frac{\alpha}{2}+R_2 \sin^2 \frac{\alpha}{2}\right)}$$

A circle $\Gamma$ and a line $\ell$ which does not intersect $\Gamma$ are given. Suppose $P, Q,R, S$ are variable points on circle $\Gamma$ such that the points $A = PQ\cap RS$ and $B = PS \cap QR$ lie on $\ell$. Prove that the circle on $AB$ as a diameter passes through two fixed points.

Given a regular hexagon, whose sidelength is $ 1$ . What is the largest number of circles of radius $\frac{\sqrt3}{4}$ can be placed without overlapping inside such a hexagon? (Circles can touch each other and the sides of the hexagon.)

Let $O$ be the circumcenter of an acute triangle $ABC$ and $R$ be its circumcenter. Consider the disks having $OA,OB,OC$ as diameters, and let $\Delta$ be the set of points in the plane belonging to at least two of the disks. Prove that the area of $\Delta$ is greater than $R^2/8$.

In the isosceles triangle $ABC$ where $AB = AC$, the point $I{}$ is the center of the inscribed circle. Through the point $A{}$ all the rays lying inside the angle $BAC$ are drawn. For each such ray, we denote by $X{}$ and $Y{}$ the points of intersection with the arc $BIC$ and the straight line $BC$ respectively. The circle $\gamma$ passing through $X{}$ and $Y{}$, which touches the arc $BIC$ at the point $X{}$ is considered. Prove that all the circles $\gamma$ pass through a fixed point.

Determine all natural numbers $n$ so that numbers $1, 2,... , n$ can be placed on the circumference of a circle and for each natural number $s$ with $1\le s \le \frac12n(n+1)$ , there is a circular arc which has the sum of all numbers in that arc to be $s$.

The projections of the body on two planes are circles. Prove that they have the same radius.

(a)Two circles $S_1$ and $S_2$ are placed so that they do not overlap each other, neither completely nor partially. They have centres in $O_1$ and $O_2$, respectively. Further, $L_1$ and $M_1$ are different points on $S_1$ so that $O_2L_1$ and $O_2M_1$ are tangent to $S_1$, and similarly $L_2$ and $M_2$ are different points on $S_2$ so that $O_1L_2$ and $O_1M_2$ are tangent to $S_2$. Show that there exists a unique circle which is tangent to the four line segments $O_2L_1, O_2M_1, O_1L_2$, and $O_1M_2$. (b) Four circles $S_1, S_2, S_3$ and $S_4$ are placed so that none of them overlap each other, neither completely nor partially. They have centres in $O_1, O_2, O_3$, and $O_4$, respectively. For each pair $(S_i, S_j )$ of circles, with $1 \le i < j \le 4$, we find a circle $S_{ij}$ as in part [b]a[/b]. The circle $S_{ij}$ has radius $R_{ij}$ . Show that $\frac{1}{R_{12}} + \frac{1}{R_{23}}+\frac{1}{R_{34}}+\frac{1}{R_{14}}= 2 \left(\frac{1}{R_{13}} +\frac{1}{R_{24}}\right)$

Pete claims that he can draw $4$ segments of length $1$ and a circle of radius less than $\sqrt3 /3 $ on a piece of paper, such that all segments would lie inside the circle and there would be no line that intersects each of $4$ segments. Is Pete right?

Let $AB$ be an arbitrary chord of the circle $(O)$. Two circles $(X)$ and $(Y )$ are on the same side of the chord $AB$ such that they are both internally tangent to $(O)$ and they are tangent to $AB$ at $C,D$ respectively, $C$ is between $A$ and $D$. Let $H$ be the intersection of $XY$ and $AB, M$ the midpoint of arc $AB$ not containing $X$ and $Y$ . Let $HM$ meet $(O)$ again at $I$. Let $IX, IY$ intersect $AB$ again at $K, J$. Prove that the circumcircle of triangle $IKJ$ is tangent to $(O)$. Nguyễn Văn Linh

Determine the radius of the largest circle which can lie on the ellipsoid $$\frac{x^2 }{a^2 } +\frac{ y^2 }{b^2 } +\frac{z^2 }{c^2 }=1 \;\;\;\; (a>b>c).$$

Let the point $ B $ lie on the circle $ S_1 $ and let the point $ A $, other than the point $ B $, lie on the tangent to the circle $ S_1 $ passing through the point $ B $. Let a point $ C $ be chosen outside the circle $ S_1 $, so that the segment $ AC $ intersects $ S_1 $ at two different points. Let the circle $ S_2 $ touch the line $ AC $ at the point $ C $ and the circle $ S_1 $ at the point $ D $, on the opposite side from the point $ B $ with respect to the line $ AC $. Prove that the center of the circumcircle of triangle $ BCD $ lies on the circumcircle of triangle $ ABC $.

Two parallel chords $AB$ and $CD$ are drawn in a circle with center $O$. Circles with diameters $AB$ and $CD$ intersect at point $P$. Prove that the midpoint of the segment $OP$ is equidistant from lines $AB$ and $CD$.

The point $M$ is the midpoint of the side $BC$ of triangle $ABC$. A circle is passing through $B$, is tangent to the line $AM$ at $M$, and intersects the segment $AB$ secondary at the point $P$. Prove that the circle, passing through $A$, $P$, and the midpoint of the segment $AM$, is tangent to the line $AC$. [i](A. Voidelevich)[/i]

Prove that the only way to cover a square of side $1$ with a finite number of circles that do not overlap, it is with only one circle of radius greater than or equal to $\frac{1}{\sqrt2}$. Circles can occupy part of the outside of the square and be of different radii.

Let $ABC$ be an acute triangle. Suppose that circle $\Gamma_1$ has it's center on the side $AC$ and is tangent to the sides $AB$ and $BC$, and circle $\Gamma_2$ has it's center on the side $AB$ and is tangent to the sides $AC$ and $BC$. The circles $\Gamma_1$ and $ \Gamma_2$ intersect at two points $P$ and $Q$. Show that if $A, P, Q$ are collinear, then $AB = AC$.