Found problems: 14
Let $AD, BE$ and $CF$ be the medians of triangle $ABC$. The points $X$ and $Y$ are the reflections of $F$ about $AD$ and $BE$, respectively. Prove that the circumcircles of triangles $BEX$ and $ADY$ are concentric.
Let $ABC$ be a scalene triangle with incenter $I$ and let $AI$ meet the circumcircle of triangle $ABC$ again at $M$. The incircle $\omega$ of triangle $ABC$ is tangent to sides $AB, AC$ at $D, E$, respectively. Let $O$ be the circumcenter of triangle $BDE$ and let $L$ be the intersection of $\omega$ and the altitude from $A$ to $BC$ so that $A$ and $L$ lie on the same side with respect to $DE$. Denote by $\Omega$ a circle centered at $O$ and passing through $L$, and let $AL$ meet $\Omega$ again at $N$.
Prove that the lines $LD$ and $MB$ meet on the circumcircle of triangle $LNE$.
Let $A_1 B_1 C_1$ and $A_2 B_2 C_2$ be two oppositely oriented concentric equilateral triangles. Prove that the lines $A_1 A_2$ , $B_1 B_2$ , and $C_1 C_2$ intersect in one point.
Let $C$ and $C'$ be two concentric circles of radii $r$ and $r'$ respectively. Determine how much the quotient $r'/r$ must be worth so that in the limited crown (annulus) through $C$ and $C'$ there are eight circles $C_i$ , $i = 1, . . . , 8$, which are tangent to $C$ and to $C'$ , and also that $C_i$ is tangent to $C_{i+1}$ for $i = 1, . . . ,7$ and $C_8$ tangent to $C_1$ .
We are given two concentric circles, Construct a square whose two vertices lie on one circle and the other two on the other circle.
Let $C_1$ and $C_2$ be two concentric circles with $C_1$ inside $C_2$. Let $P_1$ and $P_2$ be two points on $C_1$ that are not diametrically opposite. Extend the segment $P_1P_2$ past $P_2$ until it meets the circle $C_2$ in $Q_2$. The tangent to $C_2$ at $Q_2$ and the tangent to $C_1$ at $P_1$ meet in a point $X$. Draw from X the second tangent to $C_2$ which meets $C_2$ at the point $Q_1$. Show that $P_1X$ bisects angle $Q_1P_1Q_2$.
Let $ABC$ be a triangle with circumcircle $k$. The points $A_1, B_1,$ and $C_1$ on $k$ are the midpoints of arcs $\widehat{BC}$ (not containing $A$), $\widehat{AC}$ (not containing $B$), and $\widehat{AB}$ (not containing $C$), respectively. The pairwise distinct points $A_2, B_2,$ and $C_2$ are chosen such that the quadrilaterals $AB_1A_2C_1, BA_1B_2C_1,$ and $CA_1C_2B_1$ are parallelograms. Prove that $k$ and the circumcircle of triangle $A_2B_2C_2$ have a common center.
[b]Comment.[/b] Point $A_2$ can also be defined as the reflection of $A$ with respect to the midpoint of $B_1C_1$, and analogous definitions can be used for $B_2$ and $C_2$.
Given two concentric circles and a point $A$. Through point $A$, draw a secant such that its segment contained by the larger circle is divided by the smaller circle into three equal parts.
$C$, $C'$ are concentric circles with radii $R$, $R'$. A rectangle has two adjacent vertices on $C$ and the other two vertices on $C'$. Find its sides if its area is as large as possible.
Let $O$ be the center of the concentric circles $C_1,C_2$ of radii $3$ and $5$ respectively. Let $A\in C_1, B\in C_2$ and $C$ point so that triangle $ABC$ is equilateral. Find the maximum length of $ [OC] $.
Projections of two points to the sidelines of a quadrilateral lie on two concentric circles (projections of each point form a cyclic quadrilateral and the radii of circles are different). Prove that this quadrilateral is a parallelogram.
Consider (in the plane) three concentric circles with radii $1, 2$ and $3$ and equilateral triangle $\Delta$ such that on each of the three circles is one vertex of $\Delta$ . Calculate the length of the side of $\Delta$ .
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Let $M$ be the set of all tetrahedra whose inscribed and circumscribed spheres are concentric. If the radii of these spheres are denoted by $r$ and $R$ respectively, find the possible values of $R/r$ over all tetrahedra from $M$ .
Two concentric circles of radii $1$ and $2$ have centere the point $O$. The vertex $A$ of the equilateral triangle $ABC$ lies at the largest circle, while the midpoint of side $BC$ lies on the smaller circle. If$ B$,$O$ and $C$ are not collinear, what measure can the angle $\angle BOC$ have?