Let $ABCD$ be a quadrilateral and let $l$ be a line. Let $l$ intersect the lines $AB,CD,BC,DA,AC,BD$ at points $X,X',Y,Y',Z,Z'$ respectively. Given that these six points on $l$ are in the order $X,Y,Z,X',Y',Z'$, show that the circles with diameter $XX',YY',ZZ'$ are coaxal.

Let $\Gamma_1$ be a circle with center $O$ and $A$ a point on it. Consider the circle $\Gamma_2$ with center at $A$ and radius $AO$. Let $P$ and $Q$ be the intersection points of $\Gamma_1$and $\Gamma_2$. Consider the circle $\Gamma_3$ with center at $P$ and radius $PQ$. Let $C$ be the second intersection point of $\Gamma_3$ and $\Gamma_1$. The line $OP$ cuts $\Gamma_3$ at $R$ and $S$, with $R$ outside $\Gamma_1$. $RC$ cuts $\Gamma_1$ into $B$. $CS$ cuts $\Gamma_1$ into $D$. Show that $ABCD$ is a square.

Given $\vartriangle ABC$ with circumcircle $\Omega$. Assume $\omega_a, \omega_b, \omega_c$ are circles which tangent internally to $\Omega$ at $T_a,T_b, T_c $ and tangent to $BC,CA,AB$ at $P_a, P_b, P_c$, respectively. If $AT_a,BT_b,CT_c$ are collinear, prove that $AP_a,BP_b,CP_c$ are collinear.

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 circle is fixed, point $A$ is on it and point $K$ outside the circle. The secant passing through $K$ intersects circle at points $P$ and $Q$. Prove that the orthocenters of the triangle $APQ$ lie on a fixed circle.

Let $ABC$ be a triangle and $I$ its incenter. The circle passing through $A, C$ and $I$ intersect the line $BC$ for second time at point $X$. The circle passing through $B, C$ and $I$ intersects the line $AC$ for second time at point $Y$. Show that the segments $AY$ and $BX$ have equal length.

Let $ABC$ be a triangle with orthocenter $H$. Let $P$ be any point of the plane of the triangle. Let $\Omega$ be the circle with the diameter $AP$ . The circle $\Omega$ cuts $CA$ and $AB$ again at $E$ and $F$ , respectively. The line $PH$ cuts $\Omega$ again at $G$. The tangent lines to $\Omega$ at $E, F$ intersect at $T$. Let $M$ be the midpoint of $BC$ and $L$ be the point on $MG$ such that $AL$ and $MT$ are parallel. Prove that $LA$ and $LH$ are orthogonal. Lê Phúc Lữ

A circle is inscribed in a triangle and a square is circumscribed around this circle so that no side of the square is parallel to any side of the triangle. Prove that less than half of the square’s perimeter lies outside the triangle.

Consider a circle $k$ with center $S$ and radius $r$. Let a point $A\neq S$ be given with $SA=d<r$. Consider a light ray emitted at point $A$, reflected at point $B\in k$, further reflected in point $C\in k$, which then passes through the original point $A$. Compute the sinus of convex angle $SAB$ in terms of $d,r$ and discuss conditions of solvability.

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

Let $\Gamma$ be a circle and $P$ be a point outside, $PA$ and $PB$ be tangents to $\Gamma$ , $A, B \in \Gamma$ . Point $K$ is an arbitrary point on the segment $AB$. The circumscirbed circle of $\vartriangle PKB$ intersects $\Gamma$ for the second time at point $T$, point $P'$ is symmetric to point $P$ wrt point $A$. Prove that $\angle PBT = \angle P'KA$.

Two circles $\alpha,\beta$ touch externally at the point $X$. Let $A,P$ be two distinct points on $\alpha$ different from $X$, and let $AX$ and $PX$ meet $\beta$ again in the points $B$ and $Q$ respectively. Prove that $AP$ is parallel to $QB$.

On the circle of radius $30$ cm are given $2$ points A,B with $AB = 16$ cm and $C$ is a midpoint of $AB$. What is the perpendicular distance from $C$ to the circle?

$a)$ Prove that there exists $5$ nonnegative real numbers with sum equal to $1$, such that no matter how we arrange them on circle, two neighboring numbers exist with product not less than $\frac{1}{9}$ $a)$ Prove that for every $5$ nonnegative real numbers with sum equal to $1$, we can arrange them on circle, such that product of every two neighboring numbers is not greater than $\frac{1}{9}$

In the acute triangle $ABC$, the altitude $AH$ is drawn. Using segments $AB,BH,CH$ and $AC$ as diameters circles $\omega_1, \omega_2, \omega_3$ and $\omega_4$ are constructed respectively. Besides the point $H$, the circles $\omega_1$ and $\omega_3$ intersect at the point $P,$ and the circles $\omega_2$ and $\omega_4$ interext at point $Q$. The lines $BQ$ and $CP$ intersect at point $N$. Prove that this point lies on the midline of triangle $ABC$, which is parallel to $BC$.

In a given disc, construct a subset such that its area equals the half of the disc area and its intersection with its reflection over an arbitrary diameter has the area equal to the quarter of the disc area.

Let circles $\Omega$ and $\omega$ touch internally at point $A$. A chord $BC$ of $\Omega$ touches $\omega$ at point $K$. Let $O$ be the center of $\omega$. Prove that the circle $BOC$ bisects segment $AK$.

Let $\mathcal C$ be a circle centered at $O$ and $A\ne O$ be a point in its interior. The perpendicular bisector of the segment $OA$ meets $\mathcal C$ at the points $B$ and $C$, and the lines $AB$ and $AC$ meet $\mathcal C$ again at $D$ and $E$, respectively. Show that the circles $(OBC)$ and $(ADE)$ have the same centre. [i]Ion Pătrașcu, Ion Cotoi[/i]

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. ?

There are $12$ points on a circle. Four checkers, one red, one yellow, one green and one blue sit at neighboring points. In one move any checker can be moved four points to the left or right, onto the fifth point, if it is empty. If after several moves the checkers appear again at the four original points, how might their order have changed?

Consider an isosceles triangle. let $R$ be the radius of its circumscribed circle and $r$ be the radius of its inscribed circle. Prove that the distance $d$ between the centers of these two circle is \[ d=\sqrt{R(R-2r)} \]

Four right triangles, each with the sides $1$ and $2$, are assembled to a figure as shown. How large a fraction does the area of the small circle make up of that of the big one? [img]https://1.bp.blogspot.com/-XODK1XKCS0Q/XzXDtcA-xAI/AAAAAAAAMWA/zSLPpf3IcX0rgaRtOxm_F2begnVdUargACLcBGAsYHQ/s0/2010%2BMohr%2Bp1.png[/img]

On a plane, a line $\ell$ and two circles $c_1$ and $c_2$ of different radii are given such that $\ell$ touches both circles at point $P$. Point $M \ne P$ on $\ell$ is chosen so that the angle $Q_1MQ_2$ is as large as possible where $Q_1$ and $Q_2$ are the tangency points of the tangent lines drawn from $M$ to $c_i$ and $c_2$, respectively, differing from $\ell$ . Find $\angle PMQ_1 + \angle PMQ_2$·

Given a circle of radius $K$. Two other circles, the sum of the radii of which are also equal to $K$, tangent to the circle from the inside. Prove that the line connecting the points of tangency passes through one of the common points of these circles.

Circles $C_1$ and $C_2$ intersect at points $X$ and $Y$ . Point $A$ is a point on $C_1$ such that the tangent line with respect to $C_1$ passing through $A$ intersects $C_2$ at $B$ and $C$, with $A$ closer to $B$ than $C$, such that $2016 \cdot AB = BC$. Line $XY$ intersects line $AC$ at $D$. If circles $C_1$ and $C_2$ have radii of $20$ and $16$, respectively, find $\sqrt{1+BC/BD}$.