Found problems: 821
Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.
In triangle $ABC$ $\angle B = 60^o, O$ is the circumcenter, and $L$ is the foot of an angle bisector of angle $B$. The circumcirle of triangle $BOL$ meets the circumcircle of $ABC$ at point $D \ne B$. Prove that $BD \perp AC$.
(D. Shvetsov)
In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.
In a circle $S$, a chord $AB$ is drawn and let $M$ be the midpoint of the arc $AB$. Let $P$ be a point in segment $AB$ other than its midpoint. The extension of the segment $MP$ cuts $S$ in $Q$. Let $S_1$ be the circle that is tangent to the AP segments and $MP$, and also is tangent to $S$, and let $S_2$ be the circle that is tangent to the segments $BP$ and $MP$, and also tangent to $S$. The common outer tangent lines to the circles $S_1$ and $S_2$ are cut at $C$. Prove that $\angle MQC = 90^o$.
The circles $\Gamma_1$ and $\Gamma_2$ intersect at $D$ and $P$. The common tangent line of the two circles closest to point $D$ touches $\Gamma_1$ in A and $\Gamma_2$ in $B$. The line $AD$ intersects $\Gamma_2$ for the second time in $C$. Let $M$ be the midpoint of line segment $BC$. Prove that $\angle DPM = \angle BDC$.
What is the maximal number of regions a circle can be divided in by segments joining $n$ points on the boundary of the circle ?
[i]Posted already on the board I think...[/i]
Two non intersecting circles with centers $O_1$ and $O_2$ are tangent to line $s$ at points $A_1$ and $A_2$, respectively, and lying on the same side of this line. Line $O_1O_2$ intersects the first circle at $B_1$ and the second at $B_2$. Prove that the lines $A_1B_1$ and $A_2B_2$ are perpendicular to each other.
The point $M$ belongs to the side $[AC]$ of the acute-angle triangle $ABC$. Two circles are circumscribed around triangles $ABM$ and $BCM$ . What $M$ position corresponds to the minimal area of those circles intersection?
On the side $BC$ of the rectangle $ABCD$, a point $P{}$ is marked so that $\angle APD = 90^\circ$. On the straight line $AD$, points $Q{}$ and $R{}$ are selected outside the segment $AD$ such that $AQ = BP$ and $CP = DR$. The circle $\omega$ passes through the points $Q, D$ and the circumcenter of the triangle $PDQ$. The circle $\gamma$ passes through the points $A, R$ and the circumcenter of the triangle $APR$. Prove that the radius of one of the circles touching the line $AD$ and the circles $\omega$ and $\gamma$ is $2AB$.
Through the endpoints $A$ and $B$ of a diameter $AB$ of a given circle, the tangents $\ell$ and $m$ have been drawn. Let $C\ne A$ be a point on $\ell$ and let $q_1,q_2$ be two rays from $C$. Ray $q_i$ cuts the circle in $D_i$ and $E_i$ with $D_i$ between $C$ and $E_i, i = 1,2$. Rays $AD_1,AD_2,AE_1,AE_2$ meet $m$ in the respective points $M_1,M_2,N_1,N_2$. Prove that $M_1M_2 = N_1N_2$.
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 $|BD| + |AM| + |AN| = |CD| + |AP| + |AQ|$.
Is it possible to divide a circle by three chords, different from diameters, into several equal parts?
Let $ABC$ be a triangle and $P$ be a point in the plane of the triangle. The lines $AP,BP, CP$ meets $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Let $A_2,B_2,C_2$ be the Miquel point of the complete quadrilaterals $AB_1PC_1BC$, $BC_1PA_1CA$, $CA_1PB_1AB$. Prove that the circumcircles of the triangles $APA_2$,$BPB_2$, $CPC_2$, $BA_2C$, $AB_2C$, $AC_2B$ have a point of concurrency.
Nguyễn Văn Linh
Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.
Circles $\omega_1$ and $\omega_2$ with centers at points $O_1$ and $O_2$ intersect at points $A$ and $B$. A point $C$ is constructed such that $AO_2CO_1$ is a parallelogram. An arbitrary line is drawn through point $A$, which intersects the circles $\omega_1$ and $\omega_2$ for the second time at points $X$ and $Y$, respectively. Prove that $CX = CY$.
(Oleksii Masalitin)
Is it possible to place $2012$ distinct circles with the same diameter on the plane, such that each circle touches at least three others circles?
A circle $\omega$ and a point $T$ outside the circle is given. Let a tangent from $T$ to $\omega$ touch $\omega$ at $A$, and take points $B,C$ lying on $\omega$ such that $T,B,C$ are colinear. The bisector of $\angle ATC$ intersects $AB$ and $AC$ at $P$ and $Q$,respectively. Prove that $PA=\sqrt{PB\cdot QC}$
Let $\Gamma_1$ and $\Gamma_2$ be two intersecting circles with midpoints respectively $O_1$ and $O_2$, such that $\Gamma_2$ intersects the line segment $O_1O_2$ in a point $A$. The intersection points of $\Gamma_1$ and $\Gamma_2$ are $C$ and $D$. The line $AD$ intersects $\Gamma_1$ a second time in $S$. The line $CS$ intersects $O_1O_2$ in $F$. Let $\Gamma_3$ be the circumcircle of triangle $AD$. Let $E$ be the second intersection point of $\Gamma_1$ and $\Gamma_3$. Prove that $O_1E$ is tangent to $\Gamma_3$.
Two circles $\gamma_1, \gamma_2$ are given, with centers at points $O_1, O_2$ respectively. Select a point $K$ on circle $\gamma_2$ and construct two circles, one $\gamma_3$ that touches circle $\gamma_2$ at point $K$ and circle $\gamma_1$ at a point $A$, and the other $\gamma_4$ that touches circle $\gamma_2$ at point $K$ and circle $\gamma_1$ at a point $B$. Prove that, regardless of the choice of point K on circle $\gamma_2$, all lines $AB$ pass through a fixed point of the plane.
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.
One hundred points are marked inside a circle, with no three in a line. Prove that it is possible to connect the points in pairs such that all fifty lines intersect one another inside the circle.
Circles $\omega_1 , \omega_2$ intersect at points $X,Y$ and they are internally tangent to circle $\Omega$ at points $A,B$,respectively.$AB$ intersect with $\omega_1 , \omega_2$ at points $A_1,B_1$ ,respectively.Another circle is internally tangent to $\omega_1 , \omega_2$ and $A_1B_1$ at $Z$.Prove that $\angle AXZ =\angle BXZ$.(C.Ilyasov)
Is it possible to locate in a rectangle of $5$ cm by $ 8$ cm, $51$ circles of diameter $ 1$ cm, so that they don't overlap? Could it be possible for more than $40$ circles ?
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 \[ d=\sqrt{R(R-2r)} \]
The $ABCDA_1B_1C_1D_1$ cube has unit length edges. Find the distance between two circumferences, one of those is inscribed into the $ABCD$ base, and another comes through points $A,C$ and $B_1$ .