Found problems: 353
2001 Cuba MO, 4
The tangents at four different points of an arc of a circle less than $180^o$ intersect forming a convex quadrilateral $ABCD$. Prove that two of the vertices belong to an ellipse whose foci to the other two vertices.
2014 IMAC Arhimede, 2
A convex quadrilateral $ABCD$ is inscribed into a circle $\omega$ . Suppose that there is a point $X$ on the segment $AC$ such that the $XB$ and $XD$ tangents to the circle $\omega$ . Tangent of $\omega$ at $C$, intersect $XD$ at $Q$. Let $E$ ($E\ne A$) be the intersection of the line $AQ$ with $\omega$ . Prove that $AD, BE$, and $CQ$ are concurrent.
2012 Switzerland - Final Round, 3
The circles $k_1$ and $k_2$ intersect at points $D$ and $P$. The common tangent of the two circles on the side of $D$ touches $k_1$ at $A$ and $k_2$ at $B$. The straight line $AD$ intersects $k_2$ for a second time at $C$. Let $M$ be the center of the segment $BC$. Show that $ \angle DPM = \angle BDC$ .
1998 Singapore MO Open, 1
In Fig. , $PA$ and $QB$ are tangents to the circle at $A$ and $B$ respectively. The line $AB$ is extended to meet $PQ$ at $S$. Suppose that $PA = QB$. Prove that $QS = SP$.
[img]https://cdn.artofproblemsolving.com/attachments/6/f/f21c0c70b37768f3e80e9ee909ef34c57635d5.png[/img]
1949-56 Chisinau City MO, 45
Determine the locus of points, from which the tangent segments to two given circles are equal.
2009 Belarus Team Selection Test, 1
Two equal circles $S_1$ and $S_2$ meet at two different points. The line $\ell$ intersects $S_1$ at points $A,C$ and $S_2$ at points $B,D$ respectively (the order on $\ell$: $A,B,C,D$) . Define circles $\Gamma_1$ and $\Gamma_2$ as follows: both $\Gamma_1$ and $\Gamma_2$ touch $S_1$ internally and $S_2$ externally, both $\Gamma_1$ and $\Gamma_2$ line $\ell$, $\Gamma_1$ and $\Gamma_2$ lie in the different halfplanes relatively to line $\ell$. Suppose that $\Gamma_1$ and $\Gamma_2$ touch each other. Prove that $AB=CD$.
I. Voronovich
2023 Yasinsky Geometry Olympiad, 3
$ABC$ is a right triangle with $\angle C = 90^o$. Let $N$ be the middle of arc $BAC$ of the circumcircle and $K$ be the intersection point of $CN$ and $AB$. Assume $T$ is a point on a line $AK$ such that $TK=KA$. Prove that the circle with center $T$ and radius $TK$ is tangent to $BC$.
(Mykhailo Sydorenko)
2005 Sharygin Geometry Olympiad, 10.5
Two circles of radius $1$ intersect at points $X, Y$, the distance between which is also equal to $1$. From point $C$ of one circle, tangents $CA, CB$ are drawn to the other. Line $CB$ will cross the first circle a second time at point $A'$. Find the distance $AA'$.
2024 Israel TST, P1
Let $ABC$ be a triangle and let $D$ be a point on $BC$ so that $AD$ bisects the angle $\angle BAC$. The common tangents of the circles $(BAD)$, $(CAD)$ meet at the point $A'$. The points $B'$, $C'$ are defined similarly. Show that $A'$, $B'$, $C'$ are collinear.
2021 Macedonian Mathematical Olympiad, Problem 3
Let $ABCD$ be a trapezoid with $AD \parallel BC$ and $\angle BCD < \angle ABC < 90^\circ$. Let $E$ be the intersection point of the diagonals $AC$ and $BD$. The circumcircle $\omega$ of $\triangle BEC$ intersects the segment $CD$ at $X$. The lines $AX$ and $BC$ intersect at $Y$, while the lines $BX$ and $AD$ intersect at $Z$. Prove that the line $EZ$ is tangent to $\omega$ iff the line $BE$ is tangent to the circumcircle of $\triangle BXY$.
2012 Polish MO Finals, 5
Point $O$ is a center of circumcircle of acute triangle $ABC$, bisector of angle $BAC$ cuts side $BC$ in point $D$. Let $M$ be a point such that, $MC \perp BC$ and $MA \perp AD$. Lines $BM$ and $OA$ intersect in point $P$. Show that circle of center in point $P$ passing through a point $A$ is tangent to line $BC$.
2007 Greece JBMO TST, 3
Let $ABCD$ be a rectangle with $AB=a >CD =b$. Given circles $(K_1,r_1) , (K_2,r_2)$ with $r_1<r_2$ tangent externally at point $K$ and also tangent to the sides of the rectangle, circle $(K_1,r_1)$ tangent to both $AD$ and $AB$, circle $(K_2,r_2)$ tangent to both $AB$ and $BC$. Let also the internal common tangent of those circles pass through point $D$.
(i) Express sidelengths $a$ and $b$ in terms of $r_1$ and $r_2$.
(ii) Calculate the ratios $\frac{r_1}{r_2}$ and $\frac{a}{b}$ .
(iii) Find the length of $DK$ in terms of $r_1$ and $r_2$.
2024 All-Russian Olympiad Regional Round, 11.7
Graph $G_1$ of a quadratic trinomial $y = px^2 + qx + r$ with real coefficients intersects the graph $G_2$ of a quadratic trinomial $y = x^2$ in points $A$, $B$. The intersection of tangents to $G_2$ in points $A$, $B$ is point $C$. If $C \in G_1$, find all possible values of $p$.
2016 Stars of Mathematics, 3
Let $ ABC $ be a triangle, $ M_A $ be the midpoint of the side $ BC, $ and $ P_A $ be the orthogonal projection of $ A $ on $ BC. $ Similarly, define $ M_B,M_C,P_B,P_C. M_BM_C $ intersects $ P_BP_C $ at $ S_A, $ and the tangent of the circumcircle of $ ABC $ at $ A $ meets $ BC $ at $ T_A. $ Similarly, define $ S_B,S_C,T_B,T_C. $
Show that the perpendiculars through $ A,B,C, $ to $ S_AT_A,S_BT_B, $ respectively, $ S_CT_C, $ are concurent.
[i]Flavian Georgescu[/i]
Croatia MO (HMO) - geometry, 2018.3
Let $k$ be a circle centered at $O$. Let $\overline{AB}$ be a chord of that circle and $M$ its midpoint. Tangent on $k$ at points $A$ and $B$ intersect at $T$. The line $\ell$ goes through $T$, intersect the shorter arc $AB$ at the point $C$ and the longer arc $AB$ at the point $D$, so that $|BC| = |BM|$. Prove that the circumcenter of the triangle $ADM$ is the reflection of $O$ across the line $AD$
2006 Oral Moscow Geometry Olympiad, 1
The diagonals of the inscribed quadrangle $ABCD$ intersect at point $K$. Prove that the tangent at point $K$ to the circle circumscribed around the triangle $ABK$ is parallel to $CD$.
(A Zaslavsky)
2001 Croatia Team Selection Test, 2
Circles $k_1$ and $k_2$ intersect at $P$ and $Q$, and $A$ and $B$ are the tangency points of their common tangent that is closer to $P$ (where $A$ is on $k_1$ and $B$ on $k_2$). The tangent to $k_1$ at $P$ intersects $k_2$ again at $C$. The lines $AP$ and $BC$ meet at $R$. Show that the lines $BP$ and $BC$ are tangent to the circumcircle of triangle $PQR$.
2021 Adygea Teachers' Geometry Olympiad, 1
a) Two circles of radii $6$ and $24$ are tangent externally. Line $\ell$ touches the first circle at point $A$, and the second at point $B$. Find $AB$.
b) The distance between the centers $O_1$ and $O_2$ of circles of radii $6$ and $24$ is $36$. Line $\ell$ touches the first circle at point $A$, and the second at point $B$ and intersects $O_1O_2$. Find $AB$.
2021-IMOC, G7
The incircle of triangle $ABC$ tangents $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Let the tangents of $E$, $F$ with respect to $\odot(AEF)$ intersect at $P$, and $X$ be a point on $BC$ such that $EF$, $DP$, $AX$ are concurrent. Define $Q$, $Y$ and $R$, $Z$ similarly. Show that $X$, $Y$, $Z$ are collinear.
2024 Brazil National Olympiad, 4
Let \( ABC \) be an acute-angled scalene triangle. Let \( D \) be a point on the interior of segment \( BC \), different from the foot of the altitude from \( A \). The tangents from \( A \) and \( B \) to the circumcircle of triangle \( ABD \) meet at \( O_1 \), and the tangents from \( A \) and \( C \) to the circumcircle of triangle \( ACD \) meet at \( O_2 \). Show that the circle centered at \( O_1 \) passing through \( A \), the circle centered at \( O_2 \) passing through \( A \), and the line \( BC \) have a common point.
Kyiv City MO Seniors Round2 2010+ geometry, 2021.11.3.1
Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ have no common points. The line$ AB$ is a common internal tangent, and the line $CD$ is a common external tangent to these circles, where $A, C \in k_1$ and $B, D \in k_2$. Knowing that $AB=12$ and $CD =16$, find the value of the product $r_1r_2$.
2019 Saudi Arabia IMO TST, 3
Let $ABC$ be an acute nonisosceles triangle with incenter $I$ and $(d)$ is an arbitrary line tangent to $(I)$ at $K$. The lines passes through $I$, perpendicular to $IA, IB, IC$ cut $(d)$ at $A_1, B_1,C_1$ respectively. Suppose that $(d)$ cuts $BC, CA, AB$ at $M,N, P$ respectively. The lines through $M,N,P$ and respectively parallel to the internal bisectors of $A, B, C$ in triangle $ABC$ meet each other to define a triange $XYZ$. Prove that three lines $AA_1, BB_1, CC_1$ are concurrent and $IK$ is tangent to the circle $(XY Z)$
2021-IMOC, G8
Let $P$ be an arbitrary interior point of $\triangle ABC$, and $AP$, $BP$, $CP$ intersect $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Suppose that $M$ be the midpoint of $BC$, $\odot(AEF)$ and $\odot(ABC)$ intersect at $S$, $SD$ intersects $\odot(ABC)$ at $X$, and $XM$ intersects $\odot(ABC)$ at $Y$. Show that $AY$ is tangent to $\odot(AEF)$.
2022 SG Originals, Q1
For $\triangle ABC$ and its circumcircle $\omega$, draw the tangents at $B,C$ to $\omega$ meeting at $D$. Let the line $AD$ meet the circle with center $D$ and radius $DB$ at $E$ inside $\triangle ABC$. Let $F$ be the point on the extension of $EB$ and $G$ be the point on the segment $EC$ such that $\angle AFB=\angle AGE=\angle A$. Prove that the tangent at $A$ to the circumcircle of $\triangle AFG$ is parallel to $BC$.
[i]Proposed by 61plus[/i]
2020 Switzerland Team Selection Test, 8
Let $I$ be the incenter of a non-isosceles triangle $ABC$. The line $AI$ intersects the circumcircle of the triangle $ABC$ at $A$ and $D$. Let $M$ be the middle point of the arc $BAC$. The line through the point $I$ perpendicular to $AD$ intersects $BC$ at $F$. The line $MI$ intersects the circle $BIC$ at $N$.
Prove that the line $FN$ is tangent to the circle $BIC$.