Found problems: 280
2017 Costa Rica - Final Round, 5
Consider two circles $\Pi_1$ and $\Pi_1$ tangent externally at point $S$, such that the radius of $\Pi_2$ is triple the radius of $\Pi_1$. Let $\ell$ be a line that is tangent to $\Pi_1$ at point $ P$ and tangent to $\Pi_2$ at point $Q$, with $P$ and $Q$ different from $S$. Let $T$ be a point at $\Pi_2$, such that the segment $TQ$ is diameter of $\Pi_2$ and let point $R$ be the intersection of the bisector of $\angle SQT$ with $ST$. Prove that $QR = RT$.
2014 IMO, 3
Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and \[
\angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \] Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.
2017 Oral Moscow Geometry Olympiad, 4
Prove that a circle constructed with the side $AB$ of a triangle $ABC$ as a diameter touches the inscribed circle of the triangle $ABC$ if and only if the side $AB$ is equal to the radius of the exircle on that side.
1995 Czech And Slovak Olympiad IIIA, 5
Let $A,B$ be points on a circle $k$ with center $S$ such that $\angle ASB = 90^o$ . Circles $k_1$ and $k_2$ are tangent to each other at $Z$ and touch $k$ at $A$ and $B$ respectively. Circle $k_3$ inside $\angle ASB$ is internally tangent to $k$ at $C$ and externally tangent to $k_1$ and $k_2$ at $X$ and $Y$, respectively. Prove that $\angle XCY = 45^o$
XMO (China) 2-15 - geometry, 15.1
As shown in the figure, in the quadrilateral $ABCD$, $AB\perp BC$, $AD\perp CD$, let $E$ be a point on line $BD$ such that $EC = CA$. The line perpendicular on line$ AC$ passing through $E$, intersects line $AB$ at point $F$, and line $AD$ at point $G$. Let $X$ and $Y$ the midpoints of line segments $AF$ and $AG$ respectively. Let $Z$ and $W$ be the midpoints of line segments $BE$ and $DE$ respectively. Prove that the circumscribed circle of $\vartriangle WBX$ is tangent to the circumscribed circle of $\vartriangle ZDY$.
[img]https://cdn.artofproblemsolving.com/attachments/0/3/1f6fca7509e6fd6cad662b42abd236fd4858ca.jpg[/img]
2021 Mexico National Olympiad, 4
Let $ABC$ be an acutangle scalene triangle with $\angle BAC = 60^{\circ}$ and orthocenter $H$. Let $\omega_b$ be the circumference passing through $H$ and tangent to $AB$ at $B$, and $\omega_c$ the circumference passing through $H$ and tangent to $AC$ at $C$.
[list]
[*] Prove that $\omega_b$ and $\omega_c$ only have $H$ as common point.
[*] Prove that the line passing through $H$ and the circumcenter $O$ of triangle $ABC$ is a common tangent to $\omega_b$ and $\omega_c$.
[/list]
[i]Note:[/i] The orthocenter of a triangle is the intersection point of the three altitudes, whereas the circumcenter of a triangle is the center of the circumference passing through it's three vertices.
2006 Abels Math Contest (Norwegian MO), 4
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 $ABC$.
1999 ITAMO, 3
Let $r_1,r_2,r$, with $r_1 < r_2 < r$, be the radii of three circles $\Gamma_1,\Gamma_2,\Gamma$, respectively. The circles $\Gamma_1,\Gamma_2$ are internally tangent to $\Gamma$ at two distinct points $A,B$ and intersect in two distinct points. Prove that the segment $AB$ contains an intersection point of $\Gamma_1$ and $\Gamma_2$ if and only if $r_1 +r_2 = r$.
2017 Saudi Arabia Pre-TST + Training Tests, 3
Let $ABCD$ be a convex quadrilateral. Ray $AD$ meets ray $BC$ at $P$. Let $O,O'$ be the circumcenters of triangles $PCD, PAB$, respectively, $H,H'$ be the orthocenters of triangles $PCD, PAB$, respectively. Prove that circumcircle of triangle $DOC$ is tangent to circumcircle of triangle $AO'B$ if and only if circumcircle of triangle $DHC$ is tangent to circumcircle of triangle $AH'B$.
2023 Turkey MO (2nd round), 2
Let $ABC$ be a triangle and $P$ be an interior point. Let $\omega_A$ be the circle that is tangent to the circumcircle of $BPC$ at $P$ internally and tangent to the circumcircle of $ABC$ at $A_1$ internally and let $\Gamma_A$ be the circle that is tangent to the circumcircle of $BPC$ at $P$ externally and tangent to the circumcircle of $ABC$ at $A_2$ internally. Define $B_1$, $B_2$, $C_1$, $C_2$ analogously. Let $O$ be the circumcentre of $ABC$. Prove that the lines $A_1A_2$, $B_1B_2$, $C_1C_2$ and $OP$ are concurrent.
2018 India IMO Training Camp, 2
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 $\omega$.
2003 Switzerland Team Selection Test, 8
Let $A_1A_2A_3$ be a triangle and $\omega_1$ be a circle passing through $A_1$ and $A_2$.
Suppose that there are circles $\omega_2,...,\omega_7$ such that:
(a) $\omega_k$ passes through $A_k$ and $A_{k+1}$ for $k = 2,3,...,7$, where $A_i = A_{i+3}$,
(b) $\omega_k$ and $\omega_{k+1}$ are externally tangent for $k = 1,2,...,6$.
Prove that $\omega_1 = \omega_7$.
2021 Sharygin Geometry Olympiad, 9.8
A quadrilateral $ABCD$ is circumscribed around a circle $\omega$ centered at $I$. Lines $AC$ and $BD$ meet at point $P$, lines $AB$ and $CD$ meet at point $£$, lines $AD$ and $BC$ meet at point $F$. Point $K$ on the circumcircle of triangle $E1F$ is such that $\angle IKP = 90^o$. The ray $PK$ meets $\omega$ at point $Q$. Prove that the circumcircle of triangle $EQF$ touches $\omega$.
1952 Kurschak Competition, 1
A circle $C$ touches three pairwise disjoint circles whose centers are collinear and none of which contains any of the others. Show that its radius must be larger than the radius of the middle of the three circles.
2015 Sharygin Geometry Olympiad, 1
Circles $\alpha$ and $\beta$ pass through point $C$. The tangent to $\alpha$ at this point meets $\beta$ at point $B$, and the tangent to $\beta$ at $C$ meets $\alpha$ at point $A$ so that $A$ and $B$ are distinct from $C$ and angle $ACB$ is obtuse. Line $AB$ meets $\alpha$ and $\beta$ for the second time at points $N$ and $M$ respectively. Prove that $2MN < AB$.
(D. Mukhin)
Indonesia MO Shortlist - geometry, g8
$ABC$ is an acute triangle with $AB> AC$. $\Gamma_B$ is a circle that passes through $A,B$ and is tangent to $AC$ on $A$. Define similar for $ \Gamma_C$. Let $D$ be the intersection $\Gamma_B$ and $\Gamma_C$ and $M$ be the midpoint of $BC$. $AM$ cuts $\Gamma_C$ at $E$. Let $O$ be the center of the circumscibed circle of the triangle $ABC$. Prove that the circumscibed circle of the triangle $ODE$ is tangent to $\Gamma_B$.
Ukrainian TYM Qualifying - geometry, 2014.23
The inscribed circle $\omega$ of triangle $ABC$ with center $I$ touches the sides $AB, BC, CA$ at points $C_1, A_1, B_1$. The circle circumsrcibed around $\vartriangle AB_1C_1$ intersects the circumscribed circle of $ABC$ for second time at the point $K$. Let $M$ be the midpoint $BC$, $L$ be the midpoint of $B_1C_1$. The circle circumsrcibed around $\vartriangle KA_1M$ cuts intersects $\omega$ for second time at the point $T$. Prove that the circumscribed circles of triangles $KLT$ and $LIM$ are tangent.
2021 SAFEST Olympiad, 3
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
2021 Puerto Rico Team Selection Test, 5
Circle $o$ contains the circles $m$ , $p$ and $r$, such that they are tangent to $o$ internally and any two of them are tangent between themselves. The radii of the circles $m$ and $p$ are equal to $x$ . The circle $r$ has radius $1$ and passes through the center of the circle $o$. Find the value of $x$ .
2008 Dutch Mathematical Olympiad, 4
Three circles $C_1,C_2,C_3$, with radii $1, 2, 3$ respectively, are externally tangent.
In the area enclosed by these circles, there is a circle $C_4$ which is externally tangent to all three circles.
Find the radius of $C_4$.
[asy]
unitsize(0.4 cm);
pair[] O;
real[] r;
O[1] = (-12/5,16/5);
r[1] = 1;
O[2] = (0,5);
r[2] = 2;
O[3] = (0,0);
r[3] = 3;
O[4] = (-132/115, 351/115);
r[4] = 6/23;
draw(Circle(O[1],r[1]));
draw(Circle(O[2],r[2]));
draw(Circle(O[3],r[3]));
draw(Circle(O[4],r[4]));
label("$C_1$", O[1]);
label("$C_2$", O[2]);
label("$C_3$", O[3]);
[/asy]
Denmark (Mohr) - geometry, 2007.4
The figure shows a $60^o$ angle in which are placed $2007$ numbered circles (only the first three are shown in the figure). The circles are numbered according to size. The circles are tangent to the sides of the angle and to each other as shown. Circle number one has radius $1$. Determine the radius of circle number $2007$.
[img]https://1.bp.blogspot.com/-1bsLIXZpol4/Xzb-Nk6ospI/AAAAAAAAMWk/jrx1zVYKbNELTWlDQ3zL9qc_22b2IJF6QCLcBGAsYHQ/s0/2007%2BMohr%2Bp4.png[/img]
2022 Bundeswettbewerb Mathematik, 3
A circle $k$ touches a larger circle $K$ from inside in a point $P$. Let $Q$ be point on $k$ different from $P$. The line tangent to $k$ at $Q$ intersects $K$ in $A$ and $B$.
Show that the line $PQ$ bisects $\angle APB$.
2022 Portugal MO, 2
Let $P$ be a point on a circle $C_1$ and let $C_2$ be a circle with center $P$ that intersects $C_1$ at two points Q and R. The circle $C_3$, with center $Q$ and which passes through $R$, intersects $C_2$ at another point S, as in figure. Shows that $QS$ is tangent to $C_1$.
[img]https://cdn.artofproblemsolving.com/attachments/7/5/f48d414c68c33c4efaf4d6c8bebcf6f1fad4ba.png[/img]
2011 Sharygin Geometry Olympiad, 1
Altitudes $AA_1$ and $BB_1$ of triangle ABC meet in point $H$. Line $CH$ meets the semicircle with diameter $AB$, passing through $A_1, B_1$, in point $D$. Segments $AD$ and $BB_1$ meet in point $M$, segments $BD$ and $AA_1$ meet in point $N$. Prove that the circumcircles of triangles $B_1DM$ and $A_1DN$ touch.
2003 All-Russian Olympiad Regional Round, 10.6
Let $A_0$ be the midpoint of side $BC$ of triangle $ABC$, and $A'$ be the point of tangency with this side of the inscribed circle. Let's construct a circle $ \omega$ with center at $A_0$ and passing through $A'$. On other sides we will construct similar circles. Prove that if $ \omega$ is tangent to the cirucmscribed circle on arc $BC$ not containing $A$, then another one of the constructed circles touches the circumcircle.