Found problems: 333
2010 Sharygin Geometry Olympiad, 8
Triangle $ABC$ is inscribed into circle $k$. Points $A_1,B_1, C_1$ on its sides were marked, after this the triangle was erased. Prove that it can be restored uniquely if and only if $AA_1, BB_1$ and $CC_1$ concur.
Geometry Mathley 2011-12, 3.3
A triangle $ABC$ is inscribed in circle $(O)$. $P1, P2$ are two points in the plane of the triangle. $P_1A, P_1B, P_1C$ meet $(O)$ again at $A_1,B_1,C_1$ . $P_2A, P_2B, P_2C$ meet $(O)$ again at $A_2,B_2,C_2$.
a) $A_1A_2, B_1B_2, C_1C_2$ intersect $BC,CA,AB$ at $A_3,B_3,C_3$. Prove that three points $A_3,B_3,C_3$ are collinear.
b) $P$ is a point on the line $P_1P_2. A_1P,B_1P,C_1P$ meet (O) again at $A_4,B_4,C_4$. Prove that three lines $A_2A_4,B_2B_4,C_2C_4$ are concurrent.
Trần Quang Hùng
KoMaL A Problems 2019/2020, A. 779
Two circles are given in the plane, $\Omega$ and inside it $\omega$. The center of $\omega$ is $I$. $P$ is a point moving on $\Omega$. The second intersection of the tangents from $P$ to $\omega$ and circle $\Omega$ are $Q$ and $R.$ The second intersection of circle $IQR$ and lines $PI$, $PQ$ and $PR$ are $J$, $S$ and $T,$ respectively. The reflection of point $J$ across line $ST$ is $K.$
Prove that lines $PK$ are concurrent.
Estonia Open Senior - geometry, 1999.1.5
On the side $BC$ of the triangle $ABC$ a point $D$ different from $B$ and $C$ is chosen so that the bisectors of the angles $ACB$ and $ADB$ intersect on the side $AB$. Let $D'$ be the symmetrical point to $D$ with respect to the line $AB$. Prove that the points $C, A$ and $D'$ are on the same line.
2009 Tournament Of Towns, 5
Suppose that $X$ is an arbitrary point inside a tetrahedron. Through each vertex of the tetrahedron, draw a straight line that is parallel to the line segment connecting $X$ with the intersection point of the medians of the opposite face. Prove that these four lines meet at the same point.
2023 Durer Math Competition Finals, 1
$ABC$ is an isosceles triangle. The base $BC$ is $1$ cm long, and legs $AB$ and $AC$ are $2$ cm long. Let the midpoint of $AB$ be $F$, and the midpoint of $AC$ be $G$. Additionally, $k$ is a circle, that is tangent to $AB$ and A$C$, and it’s points of tangency are $F$ and $G$ accordingly. Prove, that the intersection of $CF$ and $BG$ falls on the circle $k$.
2015 IFYM, Sozopol, 6
In $\Delta ABC$ points $A_1$, $B_1$, and $C_1$ are the tangential points of the excircles of $ABC$ with its sides.
a) Prove that $AA_1$, $BB_1$, and $CC_1$ intersect in one point $N$.
b) If $AC+BC=3AB$, prove that the center of the inscribed circle of $ABC$, its tangential point with $AB$, and the point $N$ are collinear.
2006 Sharygin Geometry Olympiad, 11
In the triangle $ABC, O$ is the center of the circumscribed circle, $A ', B', C '$ are the symmetrics of $A, B, C$ with respect to opposite sides, $ A_1, B_1, C_1$ are the intersection points of the lines $OA'$ and $BC, OB'$ and $AC, OC'$ and $AB$. Prove that the lines $A A_1, BB_1, CC_1$ intersect at one point.
2006 Sharygin Geometry Olympiad, 21
On the sides $AB, BC, CA$ of triangle $ABC$, points $C', A', B'$ are taken.
Prove that for the areas of the corresponding triangles, the inequality holds:
$$S_{ABC}S^2_{A'B'C'}\ge 4S_{AB'C'}S_{BC'A'}S_{CA'B'}$$
and equality is achieved if and only if the lines $AA', BB', CC'$ intersect at one point.
2015 Switzerland - Final Round, 8
Let $ABCD$ be a trapezoid, where $AB$ and $CD$ are parallel. Let $P$ be a point on the side $BC$. Show that the parallels to $AP$ and $PD$ intersect through $C$ and $B$ to $DA$, respectively.
2003 Bosnia and Herzegovina Team Selection Test, 2
Upon sides $AB$ and $BC$ of triangle $ABC$ are constructed squares $ABB_{1}A_{1}$ and $BCC_{1}B_{2}$. Prove that lines $AC_{1}$, $CA_{1}$ and altitude from $B$ to side $AC$ are concurrent.
Swiss NMO - geometry, 2009.7
Points $A, M_1, M_2$ and $C$ are on a line in this order. Let $k_1$ the circle with center $M_1$ passing through $A$ and $k_2$ the circle with center $M_2$ passing through $C$. The two circles intersect at points $E$ and $F$. A common tangent of $k_1$ and $k_2$, touches $k_1$ at $B$ and $k_2$ at $D$. Show that the lines $AB, CD$ and $EF$ intersect at one point.
Ukrainian TYM Qualifying - geometry, 2013.17
Through the point of intersection of the medians of each of the faces a tetrahedron is drawn perpendicular to this face. Prove that all these four lines intersect at one point if and only if the four lines containing the heights of this tetrahedron intersect at one point .
Mathley 2014-15, 3
In a triangle $ABC$, $D$ is the reflection of $A$ about the sideline $BC$. A circle $(K)$ with diameter $AD$ meets $DB,DC$ at $M,N$ which are distinct from $D$. Let $E,F$ be the midpoint of $CA,AB$. The circumcircles of $KEM,KFN$ meet each other again at $L$, distinct from $K$. Let $KL$ meets $EF$ at $X$; points $Y,Z$ are defined in the same manner. Prove that three lines $AX,BY,CZ$ are concurrent.
Tran Quang Hung, Dean of the Faculty of Science, Thanh Xuan, Hanoi.
2007 Portugal MO, 2
Let $[ABC]$ be a triangle and $X, Y$ and $Z$ points on the sides $[AB], [BC]$ and $[AC]$, respectively. Prove that circumcircles of triangles $AXZ, BXY$ and $CYZ$ intersect at a point.
2011 Abels Math Contest (Norwegian MO), 2b
The diagonals $AD, BE$, and $CF$ of a convex hexagon $ABCDEF$ intersect in a common point.
Show that $a(ABE) a(CDA) a(EFC) = a(BCE) a(DEA) a(FAC)$,
where $a(KLM)$ is the area of the triangle $KLM$.
[img]https://cdn.artofproblemsolving.com/attachments/0/a/bcbbddedde159150fe3c26b1f0a2bfc322aa1a.png[/img]
2011 Sharygin Geometry Olympiad, 22
Let $CX, CY$ be the tangents from vertex $C$ of triangle $ABC$ to the circle passing through the midpoints of its sides. Prove that lines $XY , AB$ and the tangent to the circumcircle of $ABC$ at point $C$ concur.
Kyiv City MO Seniors Round2 2010+ geometry, 2013.10.3
Given a triangle $ ABC $, $ AD $ is its angle bisector. Let $ E, F $ be the centers of the circles inscribed in the triangles $ ADC $ and $ ADB $, respectively. Denote by $ \omega $, the circle circumscribed around the triangle $ DEF $, and by $ Q $, the intersection point of $ BE $ and $ CF $, and $ H, J, K, M $ , respectively the second intersection point of the lines $ CE, CF, BE, BF $ with circle $ \omega $. Let $\omega_1, \omega_2 $ the circles be circumscribed around the triangles $ HQJ $ and $ KQM $ Prove that the intersection point of the circles $\omega_1, \omega_2 $ different from $ Q $ lies on the line $ AD $.
(Kivva Bogdan)
1929 Eotvos Mathematical Competition, 3
Let $p, q$ and $r$ be three concurrent lines in the plane such that the angle between any two of them is $60^o$. Let $a$, $b$ and $c$ be real numbers such that $0 < a \le b \le c$.
(a) Prove that the set of points whose distances from $p, q$ and $r$ are respectively less than $a, b$ and $c$ consists of the interior of a hexagon if and only if $a + b > c$.
(b) Determine the length of the perimeter of this hexagon when $a + b > c$.
2017 Sharygin Geometry Olympiad, P23
Let a line $m$ touch the incircle of triangle $ABC$. The lines passing through the incenter $I$ and perpendicular to $AI, BI, CI$ meet $m$ at points $A', B', C'$ respectively. Prove that $AA', BB'$ and $CC'$ concur.
2009 Sharygin Geometry Olympiad, 3
Quadrilateral $ABCD$ is circumscribed, rays $BA$ and $CD$ intersect in point $E$, rays $BC$ and $AD$ intersect in point $F$. The incircle of the triangle formed by lines $AB, CD$ and the bisector of angle $B$, touches $AB$ in point $K$, and the incircle of the triangle formed by lines $AD, BC$ and the bisector of angle $B$, touches $BC$ in point $L$. Prove that lines $KL, AC$ and $EF$ concur.
(I.Bogdanov)
2006 Belarusian National Olympiad, 7
Let $AH_A, BH_B, CH_C$ be altitudes and $BM$ be a median of the acute-angled triangle $ABC$ ($AB > BC$). Let $K$ be a point of intersection of $BM$ and $AH_A$, $T$ be a point on $BC$ such that $KT \parallel AC$, $H$ be the orthocenter of $ABC$. Prove that the lines passing through the pairs of the points $(H_c, H_A), (H, T)$ and $(A, C)$ are concurrent.
(S. Arkhipov)
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.
2014 Contests, 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.
2017 Balkan MO Shortlist, G7
Let $ABC$ be an acute triangle with $AB\ne AC$ and circumcircle $\omega$. The angle bisector of $BAC$ intersects $BC$ and $\omega$ at $D$ and $E$ respectively. Circle with diameter $DE$ intersects $\omega$ again at $F \ne E$. Point $P$ is on $AF$ such that $PB = PC$ and $X$ and $Y$ are feet of perpendiculars from $P$ to $AB$ and $AC$ respectively. Let $H$ and $H'$ be the orthocenters of $ABC$ and $AXY$ respectively. $AH$ meets $\omega$ again at $Q$ . If $AH'$ and $HH'$ intersect the circle with diameter $AH$ again at points $S$ and $T$, respectively, prove that the lines $AT , HS$ and $FQ$ are concurrent.