Found problems: 280
Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ($\angle C=90^\circ$), $\omega_1$ and $\omega_2$ be the circumcircles of triangles $AEM$ and $CDM.$ It is known that the circles $\omega_1$ and $\omega_2$ are tangent. Find the ratio in which the circle $\omega_2$ divides $AC.$
Given triangle $ABC$. Lines $AL_a$ and $AM_a$ are the internal and the external bisectrix of angle $A$. Let $\omega_a$ be the reflection of the circumcircle of $\triangle AL_aM_a$ in the midpoint of $BC$. Circle $\omega_b$ is defined similarly. Prove that $\omega_a$ and $\omega_b$ touch if and only if $\triangle ABC$ is right-angled.
* Two circles are tangent to each other externally, and to a third one from the inside. Two common tangents to the first two circles are drawn, one outer and one inner. Prove that the inner tangent divides in halves the arc intercepted by the outer tangent on the third circle.
Circle $k_1$ has radius $10$, externally touching circle $k_2$ with radius $18$. Circle $k_3$ touches both circles, as well as the line $e$ determined by their centres. Let $k_4$ be the circle touching $k_2$ and $k_3$ externally (other than $k_1$) whose center lies on line $e$. What is the radius of $k_4$?
Let $\Gamma_1$ and $\Gamma_2$ be two circles that tangents each other at point $N$, with $\Gamma_2$ located inside $\Gamma_1$. Let $A, B, C$ be distinct points on $\Gamma_1$ such that $AB$ and $AC$ tangents $\Gamma_2$ at $D$ and $E$, respectively. Line $ND$ cuts $\Gamma_1$ again at $K$, and line $CK$ intersects line $DE$ at $I$.
(i) Prove that $CK$ is the angle bisector of $\angle ACB$.
(ii) Prove that $IECN$ and $IBDN$ are cyclic quadrilaterals.
Given are four circles $\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4$. Circles $\Gamma_1$ and $\Gamma_2$ are externally tangent at point $A$. Circles $\Gamma_2$ and
$\Gamma_3$ are externally tangent at point $B$. Circles $\Gamma_3$ and $\Gamma_4$ are externally tangent at point $C$. Circles $\Gamma_4$ and
$\Gamma_1$ are externally tangent at point $D$. Prove that $ABCD$ is cyclic.
Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other
Let $C_1$ and $C_2$ be two concentric circles and $C_3$ an outer circle to $C_1$ inner to $C_2$ and tangent to both. If the radius of $C_2$ is equal to $ 1$, how much must the radius of $C_1$ be worth, so that the area of is twice that of $C_3$?
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.
Let two circles be externally tangent at point $C$, with parallel diameters $A_1A_2, B_1B_2$ (i.e. the quadrilateral $A_1B_1B_2A_2$ is a trapezoid with bases $A_1A_2$ and $B_1B_2$ or parallelogram). Circle with the center on the common internal tangent to these two circles, passes through the intersection point of lines $A_1B_2$ and $A_2B_1$ as well intersects those lines at points $M, N$. Prove that the line $MN$ is perpendicular to the parallel diameters $A_1A_2, B_1B_2$.
(Yuri Biletsky)
As shown in the figure, $\odot O$ is the circumcircle of $\vartriangle ABC$, $\odot J$ is inscribed in $\odot O$ and is tangent to $AB$, $AC$ at points $D$ and E respectively, line segment $FG$ and $\odot O$ are tangent to point $A$, and $AF =AG=AD$, the circumscribed circle of $\vartriangle AFB$ intersects $\odot J$ at point $S$. Prove that the circumscribed circle of $\vartriangle ASG$ is tangent to $\odot J$.
[img]https://cdn.artofproblemsolving.com/attachments/a/a/62d44e071ea9903ebdd68b43943ba1d93b4138.png[/img]
A semicircle of radius $5$ and a quarter of a circle of radius $8$ touch each other and are located inside the square as shown in the figure. Find the length of the part of the common tangent, enclosed in the same square.
[img]https://cdn.artofproblemsolving.com/attachments/f/2/010f501a7bc1d34561f2fe585773816f168e93.png[/img]
Consider a semicircle of center $O$ and diameter $[AB]$, and let $C$ be an arbitrary point on the segment $(OB)$. The perpendicular to the line $AB$ through $C$ intersects the semicircle in $D$. A circle centered in $P$ is tangent to the arc $BD$ in $F$ and to the segments $[AB]$ and $[CD]$ in $G$ and $E$, respectively. Prove that the triangle $ADG$ is isosceles.
Let $ABC$ be acute-angled triangle with circumcircle $\Gamma$. Points $H$ and $M$ are the orthocenter and the midpoint of $BC$ respectively. The line $HM$ meets the circumcircle $\omega$ of triangle $BHC$ at point $N\not= H$. Point $P$ lies on the arc $BC$ of $\omega$ not containing $H$ in such a way that $\angle HMP = 90^\circ$. The segment $PM$ meets $\Gamma$ at point $Q$. Points $B'$ and $C'$ are the reflections of $A$ about $B$ and $C$ respectively. Prove that the circumcircles of triangles $AB'C'$ and $PQN$ are tangent.
Let $\gamma$ be the circumscribed circle about a right-angled triangle $ABC$ with right angle $C$. Let $\delta$ be the circle tangent to the sides $AC$ and $BC$ and tangent to the circle $\gamma$ internally.
(a) Find the radius $i$ of $\delta$ in terms of $a$ when $AC$ and $BC$ both have length $a$.
(b) Show that the radius $i$ is twice the radius of the inscribed circle of $ABC$.
$\vartriangle ABC$ is a scalene triangle with circumcircle $\Omega$. For a arbitrary $X$ in the plane, define $D_x,E_x, F_x$ to be the intersection of tangent line of $X$ (with respect to $BXC$) and $BC,CA,AB$, respectively. Let the intersection of $AX$ with $\Omega$ be $S_x$ and $T_x = D_xS_x \cap \Omega$. Show that $\Omega$ and circumcircle of $\vartriangle T_xE_xF_x$ are tangent to each other.
[img]https://2.bp.blogspot.com/-rTMODHbs5Ac/XnYNQYjYzBI/AAAAAAAALeg/576nGDQ6NDA0-W5XqiNczNtI07cEZxPeQCK4BGAYYCw/s1600/imoc2019g4.png[/img]
Let $ABC$ be a triangle such that $\measuredangle BAC = 60^{\circ}$. Let $D$ and $E$ be the feet of the perpendicular from $A$ to the bisectors of the external angles of $B$ and $C$ in triangle $ABC$, respectively. Let $O$ be the circumcenter of the triangle $ABC$. Prove that circumcircle of the triangle $BOC$ has exactly one point in common with the circumcircle of $ADE$.
Two circles $k_1,k_2$ touch each other at $P$, and touch a line $\ell$ at $A$ and $B$ respectively.
Line $AP$ meets $k_2$ at $C$. Prove that $BC$ is perpendicular to $\ell$.
Let $ABC$ be an acute triangle, with $AB <AC$. Let $M$ be the midpoint of $AB$, $H$ the foot of the altitude from $A$, and $Q$ be point on side $AC$ such that $\angle ABQ = \angle BCA$. Show that the circumcircles of the triangles $ABQ$ and $BHM$ are tangent.
Three circles are pairwise tangent, with none of them lying inside another. The centres of the circles are the corners of a triangle with circumference $1$. What is the smallest possible value for the sum of the areas of the circles?
Let $\Gamma_1$ and $\Gamma_2$ be circles internally tangent at point $A$, with $\Gamma_1$ inside $\Gamma_2$. Let $BC$ be a chord of $\Gamma_2$ which is tangent to $\Gamma_1$ at point $D$. Prove that line $AD$ is the angle bisector of $\angle BAC$.
In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.
Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.
Let $ABC$ be a triangle and let the points $D, E$ be on the rays $AB$, $AC$ such that $BCED$ is cyclic. Prove that the following two statements are equivalent:
$\bullet$ There is a point $X$ on the circumcircle of $ABC$ such that $BDX$, $CEX$ are tangent to each other.
$\bullet$ $AB \cdot AD \le 4R^2$, where $R$ is the radius of the circumcircle of $ABC$.
In the picture below, the circles are tangent to each other and to the edges of the rectangle. The larger circle's radius equals 1. Determine the area of the rectangle.
[img]https://i.imgur.com/g3GUg4Z.png[/img]