Found problems: 821
There are four circles. The chord$ AB$ is drawn in the first one, and the distance from the midpoint of the smaller of the two formed arcs to $AB$ is equal to $1$. The second, third and fourth circles are located inside the larger segment and touch the chord $AB$. The second and fourth circles touch internally the first and externally the third. The sum of the radii of the last three circles is equal to the radius of the first circle. Find the radius of the third circle if it is known that the line passing through the centers of the first and third circles is not parallel to the line passing through the centers of the other two circles.
Let $A$ and $B$ be two points inside a given circle $k$. Prove that there exist (infinitely many) circles through $A$ and $B$ which lie entirely in $k$.
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 $AED$.
Given a circle with center $O$ and radius equal to $1$. $AB$ and $AC$ are the tangents to this circle from point $A$. Point $M$ on the circle is such that the areas of quadrilaterals $OBMC$ and $ABMC$ are equal. Find $MA$.
On the circle with diameter $AB$, the point $M$ was selected and fixed. Then the point ${{Q} _ {i}}$ is selected, for which the chord $M {{Q} _ {i}}$ intersects $AB$ at the point ${{K} _ {i}}$ and thus $ \angle M {{K} _ {i}} B <90 {} ^ \circ$. A chord that is perpendicular to $AB$ and passes through the point ${{K} _ {i}}$ intersects the line $B {{Q} _ {i}}$ at the point ${{P } _ {i}}$. Prove that the points ${{P} _ {i}}$ in all possible choices of the point ${{Q} _ {i}}$ lie on the same line.
(Igor Nagel)
Nine circles are drawn around an arbitrary triangle as in the figure. All circles tangent to the same side of the triangle have equal radii. Three lines are drawn, each one connecting one of the triangle’s vertices to the center of one of the circles touching the opposite side, as in the figure. Show that the three lines are concurrent.
(N. Beluhov)
A quadrilateral $ABCD$ with perpendicular diagonals is inscribed into a circle $\omega$. Two arcs $\alpha$ and $\beta$ with diameters AB and $CD$ lie outside $\omega$. Consider two crescents formed by the circle $\omega$ and the arcs $\alpha$ and $\beta$ (see Figure). Prove that the maximal radii of the circles inscribed into these crescents are equal.
(F.Nilov)
Three circles in a plane have the sides of a triangle as their diameters. Prove that there is a point that is in the interior of all three circles.
In the plane are given a circle $k$ with radii $R$ and the points $A_1,A_2,\ldots,A_n$, lying on $k$ or outside $k$. Prove that there exist infinitely many points $X$ from the given circumference for which
$$\sum_{i=1}^n A_iX^2\ge2nR^2.$$
Does there exist a pair of points on different sides of some diameter, $X$ and $Y$ from $k$, such that
$$\sum_{i=1}^n A_iX^2\ge2nR^2\text{ and }\sum_{i=1}^n A_iY^2\ge2nR^2?$$
[i]H. Lesov[/i]
Given are two circles which intersect at points $P$ and $Q$. Consider an arbitrary line $\ell$ through $Q$, let the second points of intersection of this line with the circles be $A$ and $B$ respectively. Let $C$ be the point of intersection of the tangents to the circles in those points. Let $D$ be the intersection of the line $AB$ and the bisector of the angle $CPQ$. Prove that all possible $D$ for any choice of $\ell$ lie on a single circle.
Alexey Zaslavsky
Quadrilateral$ ABCD$ has two diagonals $AC,BD$ that are mutually perpendicular. Let $M$ be the Miquel point of the complete quadrilateral formed by lines $AB,BC,CD,DA$. Suppose that $L$ is the intersection of two circles $(MAC)$ and $(MBD)$. Prove that the circumcenters of triangles $LAB,LBC,LCD,LDA$ are on the same circle called $\omega$ and that three circles $(MAC), (MBD), \omega$ are pairwise orthogonal.
Nguyễn Văn Linh
On the circle with center 0 and radius 1 the point $A_0$ is fixed and points $A_1, A_2, \ldots, A_{999}, A_{1000}$ are distributed in such a way that the angle $\angle A_00A_k = k$ (in radians). Cut the circle at points $A_0, A_1, \ldots, A_{1000}.$ How many arcs with different lengths are obtained. ?
Two circles $\omega_1$ and $\omega_2$ intersecting at points $X{}$ and $Y{}$ are inside the circle $\Omega$ and touch it at points $A{}$ and $B{}$, respectively; the segments $AB$ and $XY$ intersect. The line $AB$ intersects the circles $\omega_1$ and $\omega_2$ again at points $C{}$ and $D{}$, respectively. The circle inscribed in the curved triangle $CDX$ touches the side $CD$ at the point $Z{}$. Prove that $XZ$ is a bisector of $\angle AXB{}$.
Three circles are pair-wise tangent to each other. Prove that the circle passing through the three tangent points is perpendicular to each of the initial three circles.
We have three circles $w_1$, $w_2$ and $\Gamma$ at the same side of line $l$ such that $w_1$ and $w_2$ are tangent to $l$ at $K$ and $L$ and to $\Gamma$ at $M$ and $N$, respectively. We know that $w_1$ and $w_2$ do not intersect and they are not in the same size. A circle passing through $K$ and $L$ intersect $\Gamma$ at $A$ and $B$. Let $R$ and $S$ be the reflections of $M$ and $N$ with respect to $l$. Prove that $A, B, R, S$ are concyclic.
a) What is the least number of points that can be chosen on a circle of length $1956$, so that for each of these points there is exactly one chosen point at distance $1$, and exactly one chosen point at distance $2$ (distances are measured along the circle)?
b) On a circle of length $15$ there are selected $n$ points such that for each of them there is exactly one selected point at distance $1$ from it, and exactly one is selected point at distance $2$ from it. (All distances are measured along the circle.) Prove that $n$ is divisible by $10$.
Circles ${{w} _ {1}}$ and ${{w} _ {2}}$ with centers ${{O} _ {1}}$ and ${{O} _ {2}}$ intersect at points $A$ and $B$, respectively. The line ${{O} _ {1}} {{O} _ {2}}$ intersects ${{w} _ {1}}$ at the point $Q$, which does not lie inside the circle ${{w} _ {2}}$, and ${{w} _ {2}}$ at the point $X$ lying inside the circle ${{w} _ {1} }$. Around the triangle ${{O} _ {1}} AX$ circumscribe a circle ${{w} _ {3}}$ intersecting the circle ${{w} _ {1}}$ for the second time in point $T$. The line $QT$ intersects the circle ${{w} _ {3}}$ at the point $K$, and the line $QB$ intersects ${{w} _ {2}}$ the second time at the point $H$. Prove that
a) points $T, \, \, X, \, \, B$ lie on one line;
b) points $K, \, \, X, \, \, H$ lie on one line.
(Vadym Mitrofanov)
Let $ABC$ be an acute triangle, and its altitudes $AX,BY,CZ$ concurrent at $H$. Construct circles $(K_a), (K_b), (K_c)$ circumscribing the triangles $AY Z, BZX, CXY$ . Construct a circle $(K)$ that is internally tangent to all the three circles $(Ka), (K_b), (K_c)$. Prove that $(K)$ is tangent to the circumcircle $(O)$ of the triangle $ABC$.
Đỗ Thanh Sơn
In a circle of radius $1$, six arcs of radius $1$ are drawn, which cut the circle as in the figure. Determine the black area.
[img]https://cdn.artofproblemsolving.com/attachments/8/9/0323935be8406ea0c452b3c8417a8148c977e3.jpg[/img]
Let $C_1,C_2,..., C_n$ , $(n\ge 3)$ be circles having a common point $M$. Three straight lines passing through $M$ intersect again the circles in $A_1, A_2,..., A_n$ ; $B_1,B_2,..., B_n$ and $X_1,X_2,..., X_n$ respectively. Prove that if
$$A_1A_2 =A_2A_3 =...=A_{n-1}A_n$$ and $$B_1B_2 =B_2B_3 =...=B_{n-1}B_n$$ then $$X_1X_2 =X_2X_3 =...=X_{n-1}X_n.$$
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 $\beta$.
Let $g_1$ and $g_2$ be some lines, which intersect in point $A$. A circle $k_1$ is tangent to $g_1$ at point $A$ and intersects $g_2$ for a second time in $C$. A circle $k_2$ is tangent to $g_2$ at point $A$ and intersects $g_1$ for a second time in $D$. The circles $k_1$ and $k_2$ intersect for a second time in point $B$. Prove that, if $\frac{AC}{AD}=\sqrt{2}$, then $\frac{BC}{BD}=2$.
Let $\omega$ be the semicircle with diameter $PQ$. A circle $k$ is tangent internally to $\omega$ and to segment $PQ$ at $C$. Let $AB$ be the tangent to $K$ perpendicular to $PQ$, with $A$ on $\omega$ and $B$ on the segment $CQ$. Show that $AC$ bisects angle $\angle PAB$
Rays $l$ and $m$ forming an angle of $a$ are drawn from the same point. Let $P$ be a fixed point on $l$. For each circle $C$ tangent to $l$ at $P$ and intersecting $m$ at $Q$ and $R$, let $T$ be the intersection point of the bisector of angle $QPR$ with $C$. Describe the locus of $T$ and justify your answer.
Fix a triangle $ABC$. Let $\Gamma_1$ the circle through $B$, tangent to edge in $A$. Let $\Gamma_2$ the circle through C tangent to edge $AB$ in $A$. The second intersection of $\Gamma_1$ and $\Gamma_2$ is denoted by $D$. The line $AD$ has second intersection $E$ with the circumcircle of $\vartriangle ABC$. Show that $D$ is the midpoint of the segment $AE$.