Found problems: 353
2019 Brazil Team Selection Test, 2
Let $ABC$ be a triangle, and $A_1$, $B_1$, $C_1$ points on the sides $BC$, $CA$, $AB$, respectively, such that the triangle $A_1B_1C_1$ is equilateral. Let $I_1$ and $\omega_1$ be the incenter and the incircle of $AB_1C_1$. Define $I_2$, $\omega_2$ and $I_3$, $\omega_3$ similarly, with respect to the triangles $BA_1C_1$ and $CA_1B_1$, respectively. Let $l_1 \neq BC$ be the external tangent line to $\omega_2$ and $\omega_3$. Define $l_2$ and $l_3$ similarly, with respect to the pairs $\omega_1$, $\omega_3$ and $\omega_1$, $\omega_2$.
Knowing that $A_1I_2 = A_1I_3$, show that the lines $l_1$, $l_2$, $l_3$ are concurrent.
2017 Yasinsky Geometry Olympiad, 3
Given circle arc, whose center is an inaccessible point. $A$ is a point on this arc (see fig.). How to construct using compass and ruler without divisions, a tangent to given circle arc at point $A$ ?
[img]https://1.bp.blogspot.com/-7oQBNJGLsVw/W6dYm4Xw7bI/AAAAAAAAJH8/sJ-rgAQZkW0kvlPOPwYiGjnOXGQZuDnRgCK4BGAYYCw/s1600/Yasinsky%2B2017%2BVIII-IX%2Bp3.png[/img]
Indonesia MO Shortlist - geometry, g5
Let $ABC$ be an acute triangle. Suppose that circle $\Gamma_1$ has it's center on the side $AC$ and is tangent to the sides $AB$ and $BC$, and circle $\Gamma_2$ has it's center on the side $AB$ and is tangent to the sides $AC$ and $BC$. The circles $\Gamma_1$ and $ \Gamma_2$ intersect at two points $P$ and $Q$. Show that if $A, P, Q$ are collinear, then $AB = AC$.
2008 Balkan MO Shortlist, G6
On triangle $ABC$ the $AM$ ($M\in BC$) is median and $BB_1$ and $CC_1$ ($B_1 \in AC,C_1 \in AB$) are altitudes. The stright line $d$ is perpendicular to $AM$ at the point $A$ and intersect the lines $BB_1$ and $CC_1$ at the points $E$ and $F$ respectively. Let denoted with $\omega$ the circle passing through the points $E, M$ and $F$ and with $\omega_1$ and with $\omega_2$ the circles that are tangent to segment $EF$ and with $\omega$ at the arc $EF$ which is not contain the point $M$. If the points $P$ and $Q$ are intersections points for $\omega_1$ and $\omega_2$ then prove that the points $P, Q$ and $M$ are collinear.
2015 Indonesia MO Shortlist, G3
Given $ABC$ triangle with incircle $L_1$ and circumcircle $L_2$. If points $X, Y, Z$ lie on $L_2$, such that $XY, XZ$ are tangent to $L_1$, then prove that $YZ$ is also tangent to $L_1$.
1952 Moscow Mathematical Olympiad, 225
From a point $C$, tangents $CA$ and $CB$ are drawn to a circle $O$. From an arbitrary point $N$ on the circle, perpendiculars $ND, NE, NF$ are drawn on $AB, CA$ and $CB$, respectively. Prove that the length of $ND$ is the mean proportional of the lengths of $NE$ and $NF$.
1996 Singapore Senior Math Olympiad, 1
$PQ, CD$ are parallel chords of a circle. The tangent at $D$ cuts $PQ$ at $T$ and $B$ is the point of contact of the other tangent from $T$ (Fig. ). Prove that $BC$ bisects $PQ$.
[img]https://cdn.artofproblemsolving.com/attachments/2/f/22f69c03601fbb8e388e319cd93567246b705c.png[/img]
2023 Indonesia MO, 7
Given a triangle $ABC$ with $\angle ACB = 90^{\circ}$. Let $\omega$ be the circumcircle of triangle $ABC$. The tangents of $\omega$ at $B$ and $C$ intersect at $P$. Let $M$ be the midpoint of $PB$. Line $CM$ intersects $\omega$ at $N$ and line $PN$ intersects $AB$ at $E$. Point $D$ is on $CM$ such that $ED \parallel BM$. Show that the circumcircle of $CDE$ is tangent to $\omega$.
2022 Azerbaijan EGMO/CMO TST, G1
Let $ABC$ be an isosceles triangle with $AC = BC$ and circumcircle $k$. The point $D$ lies on the shorter arc of $k$ over the chord $BC$ and is different from $B$ and $C$. Let $E$ denote the intersection of $CD$ and $AB$. Prove that the line through $B$ and $C$ is a tangent of the circumcircle of the triangle $BDE$.
(Karl Czakler)
2015 Romania Team Selection Tests, 1
Two circles $\gamma $ and $\gamma'$ cross one another at points $A$ and $B$ . The tangent to $\gamma'$ at $A$ meets $\gamma$ again at $C$ , the tangent to $\gamma$ at $A$ meets $\gamma'$ again at $C'$ , and the line $CC'$ separates the points $A$ and $B$ . Let $\Gamma$ be the circle externally tangent to $\gamma$ , externally tangent to $\gamma'$ , tangent to the line $CC'$, and lying on the same side of $CC'$ as $B$ . Show that the circles $\gamma$ and $\gamma'$ intercept equal segments on one of the tangents to $\Gamma$ through $A$ .
III Soros Olympiad 1996 - 97 (Russia), 9.10
Let $M$ be the intersection point of the diagonals of the parallelogram $ABCD$. Consider three circles passing through $M$, the first and second touch $AB$ at points $A$ and $B$, respectively, and the third passes through $C$ and $D$. Let us denote by $P$ and $C$, respectively, the intersection points of the first circle with the third and the second with the third, different from $M$. Prove that the line $PQ$ touches the first and second circles.
2017 Sharygin Geometry Olympiad, P16
The tangents to the circumcircle of triangle $ABC$ at $A$ and $B$ meet at point $D$. The circle passing through the projections of $D$ to $BC, CA, AB$, meet $AB$ for the second time at point $C'$. Points $A', B'$ are defined similarly. Prove that $AA', BB', CC'$ concur.
2015 EGMO, 1
Let $\triangle ABC$ be an acute-angled triangle, and let $D$ be the foot of the altitude from $C.$ The angle bisector of $\angle ABC$ intersects $CD$ at $E$ and meets the circumcircle $\omega$ of triangle $\triangle ADE$ again at $F.$
If $\angle ADF = 45^{\circ}$, show that $CF$ is tangent to $\omega .$
2010 Estonia Team Selection Test, 4
In an acute triangle $ABC$ the angle $C$ is greater than the angle $A$. Let $AE$ be a diameter of the circumcircle of the triangle. Let the intersection point of the ray $AC$ and the tangent of the circumcircle through the vertex $B$ be $K$. The perpendicular to $AE$ through $K$ intersects the circumcircle of the triangle $BCK$ for the second time at point $D$. Prove that $CE$ bisects the angle $BCD$.
2013 Saudi Arabia IMO TST, 1
Triangle $ABC$ is inscribed in circle $\omega$. Point $P$ lies inside triangle $ABC$.Lines $AP,BP$ and $CP$ intersect $\omega$ again at points $A_1$, $B_1$ and $C_1$ (other than $A, B, C$), respectively. The tangent lines to $\omega$ at $A_1$ and $B_1$ intersect at $C_2$.The tangent lines to $\omega$ at $B_1$ and $C_1$ intersect at $A_2$. The tangent lines to $\omega$ at $C_1$ and $A_1$ intersect at $B_2$. Prove that the lines $AA_2,BB_2$ and $CC_2$ are concurrent.
2017 Ukrainian Geometry Olympiad, 3
On the hypotenuse $AB$ of a right triangle $ABC$, we denote a point $K$ such that $BK = BC$. Let $P$ be a point on the perpendicular from the point $K$ to line $CK$, equidistant from the points $K$ and $B$. Let $L$ be the midpoint of $CK$. Prove that line $AP$ is tangent to the circumcircle of $\Delta BLP$.
1995 All-Russian Olympiad Regional Round, 10.3
In an acute-angled triangle $ABC$, the circle $S$ with the altitude $BK$ as the diameter intersects $AB$ at $E$ and $BC$ at $F$. Prove that the tangents to $S$ at $E$ and $F$ meet on the median from $B$.
2022 Iran Team Selection Test, 8
In triangle $ABC$, with $AB<AC$, $I$ is the incenter, $E$ is the intersection of $A$-excircle and $BC$. Point $F$ lies on the external angle bisector of $BAC$ such that $E$ and $F$ lieas on the same side of the line $AI$ and $\angle AIF=\angle AEB$. Point $Q$ lies on $BC$ such that $\angle AIQ=90$. Circle $\omega_b$ is tangent to $FQ$ and $AB$ at $B$, circle $\omega_c$ is tangent to $FQ$ and $AC$ at $C$ and both circles pass through the inside of triangle $ABC$. if $M$ is the Midpoint od the arc $BC$, which does not contain $A$, prove that $M$ lies on the radical axis of $\omega_b$ and $\omega_c$.
Proposed by Amirmahdi Mohseni
1998 Bosnia and Herzegovina Team Selection Test, 4
Circle $k$ with radius $r$ touches the line $p$ in point $A$. Let $AB$ be a dimeter of circle and $C$ an arbitrary point of circle distinct from points $A$ and $B$. Let $D$ be a foot of perpendicular from point $C$ to line $AB$. Let $E$ be a point on extension of line $CD$, over point $D$, such that $ED=BC$. Let tangents on circle from point $E$ intersect line $p$ in points $K$ and $N$. Prove that length of $KN$ does not depend from $C$
2013 Korea Junior Math Olympiad, 2
A pentagon $ABCDE$ is inscribed in a circle $O$, and satis es $AB = BC , AE = DE$. The circle that is tangent to $DE$ at $E$ and passing $A$ hits $EC$ at $F$ and $BF$ at $G (\ne F)$. Let $DG\cap O = H (\ne D)$. Prove that the tangent to $O$ at $E$ is perpendicular to $HA$.
2018 Dutch IMO TST, 4
In a non-isosceles triangle $ABC$ the centre of the incircle is denoted by $I$. The other intersection point of the angle bisector of $\angle BAC$ and the circumcircle of $\vartriangle ABC$ is $D$. The line through $I$ perpendicular to $AD$ intersects $BC$ in $F$. The midpoint of the circle arc $BC$ on which $A$ lies, is denoted by $M$. The other intersection point of the line $MI$ and the circle through $B, I$ and $C$, is denoted by $N$. Prove that $FN$ is tangent to the circle through $B, I$ and $C$.
1969 IMO, 4
$C$ is a point on the semicircle diameter $AB$, between $A$ and $B$. $D$ is the foot of the perpendicular from $C$ to $AB$. The circle $K_1$ is the incircle of $ABC$, the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$.
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 $BIC$.
2013 German National Olympiad, 3
Given two circles $k_1$ and $k_2$ which intersect at $Q$ and $Q'.$ Let $P$ be a point on $k_2$ and inside of $k_1 $ such that the line $PQ$ intersects $k_1$ in a point $X\ne Q$ and such that the tangent to $k_1$ at $X$ intersects $k_2$ in points $A$ and $B.$ Let $k$ be the circle through $A,B$ which is tangent to the line through $P$ parallel to $AB.$
Prove that the circles $k_1$ and $k$ are tangent.
2017 Peru IMO TST, 3
The inscribed circle of the triangle $ABC$ is tangent to the sides $BC, AC$ and $AB$ at points $D, E$ and $F$, respectively. Let $M$ be the midpoint of $EF$. The circle circumscribed around the triangle $DMF$ intersects line $AB$ at $L$, the circle circumscribed around the triangle $DME$ intersects the line $AC$ at $K$. Prove that the circle circumscribed around the triangle $AKL$ is tangent to the line $BC$.