Found problems: 333
2019 Thailand TSTST, 3
Let $ABC$ be an acute triangle with $AX, BY$ and $CZ$ as its altitudes.
$\bullet$ Line $\ell_A$, which is parallel to $YZ$, intersects $CA$ at $A_1$ between $C$ and $A$, and intersects $AB$ at $A_2$ between $A$ and $B$.
$\bullet$ Line $\ell_B$, which is parallel to $ZX$, intersects $AB$ at $B_1$ between $A$ and $B$, and intersects $BC$ at $B_2$ between $B$ and $C$.
$\bullet$ Line $\ell_C$, which is parallel to $XY$ , intersects $BC$ at $C_1$ between $B$ and $C$, and intersects $CA$ at $C_2$ between $C$ and $A$.
Suppose that the perimeters of the triangles $\vartriangle AA_1A_2$, $\vartriangle BB_1B_2$ and $\vartriangle CC_1C_2$ are equal to $CA+AB,AB +BC$ and $BC +CA$, respectively. Prove that $\ell_A, \ell_B$ and $\ell_C$ are concurrent.
2020 Federal Competition For Advanced Students, P2, 1
Let $ABCD$ be a convex cyclic quadrilateral with the diagonal intersection $S$. Let further be $P$ the circumcenter of the triangle $ABS$ and $Q$ the circumcenter of the triangle $BCS$. The parallel to $AD$ through $P$ and the parallel to $CD$ through $Q$ intersect at point $R$. Prove that $R$ is on $BD$.
(Karl Czakler)
2015 Cuba MO, 6
Let $ABC$ be a triangle such that $AB > AC$, with a circumcircle $\omega$. Draw the tangents to $\omega$ at $B$ and $C$ and these intersect at $P$. The perpendicular to $AP$ through $A$ cuts $BC$ at $R$. Let $S$ be a point on the segment $PR$ such that $PS = PC$.
(a) Prove that the lines $CS$ and $AR$ intersect on $\omega$.
(b) Let $M$ be the midpoint of $BC$ and $Q$ be the point of intersection of $CS$ and $AR$. Circle $\omega$ and the circumcircle of $\vartriangle AMP$ intersect at a point $J$ ($J \ne A$), prove that $P$, $J$ and $Q$ are collinear.
1951 Kurschak Competition, 1
$ABCD$ is a square. $E$ is a point on the side $BC$ such that $BE =1/3 BC$, and $F$ is a point on the ray $DC$ such that $CF =1/2 DC$. Prove that the lines $AE$ and $BF$ intersect on the circumcircle of the square.
[img]https://cdn.artofproblemsolving.com/attachments/e/d/09a8235d0748ce4479e21a3bb09b0359de54b5.png[/img]
Ukrainian TYM Qualifying - geometry, 2012.11
Let $E$ be an arbitrary point on the side $BC$ of the square $ABCD$. Prove that the inscribed circles of triangles $ABE$, $CDE$, $ADE$ have a common tangent.
2008 Switzerland - Final Round, 8
Let $ABCDEF$ be a convex hexagon inscribed in a circle . Prove that the diagonals $AD, BE$ and $CF$ intersect at one point if and only if $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$
2009 Junior Balkan Team Selection Tests - Romania, 3
Consider a regular polygon $A_0A_1...A_{n-1}, n \ge 3$, and $m \in\{1, 2, ..., n - 1\}, m \ne n/2$. For any number $i \in \{0,1, ... , n - 1\}$, let $r(i)$ be the remainder of $i + m$ at the division by $n$. Prove that no three segments $A_iA_{r(i)}$ are concurrent.
Geometry Mathley 2011-12, 4.4
Let $ABC$ be a triangle with $E$ being the centre of its Euler circle. Through $E$, construct the lines $PS, MQ, NR$ parallel to $BC,CA,AB$ ($R,Q$ are on the line $BC, N, P$ on the line $AC,M, S$ on the line $AB$). Prove that the four Euler lines of triangles $ABC,AMN,BSR,CPQ$ are concurrent.
Nguyễn Văn Linh
2019 Saudi Arabia Pre-TST + Training Tests, 3.1
Let $ABC$ be a triangle inscribed in a circle ($\omega$) and $I$ is the incenter. Denote $D,E$ as the intersection of $AI,BI$ with ($\omega$). And $DE$ cuts $AC,BC$ at $F,G$ respectively. Let $P$ be a point such that $PF \parallel AD$ and $PG \parallel BE$. Suppose that the tangent lines of ($\omega$) at $A,B$ meet at $K$. Prove that three lines $AE,BD,KP$ are concurrent or parallel.
2008 Austria Beginners' Competition, 4
Let $ABC$ be an acute-angled triangle with the property that the bisector of $\angle BAC$, the altitude through $B$ and the perpendicular bisector of $AB$ intersect in one point. Determine the angle $\alpha = \angle BAC$.
1961 Kurschak Competition, 3
Two circles centers $O$ and $O'$ are disjoint. $PP'$ is an outer tangent (with $P$ on the circle center O, and P' on the circle center $O'$). Similarly, $QQ'$ is an inner tangent (with $Q$ on the circle center $O$, and $Q'$ on the circle center $O'$). Show that the lines $PQ$ and $P'Q'$ meet on the line $OO'$.
[img]https://cdn.artofproblemsolving.com/attachments/b/d/bad305631571323a61b097f149a1bb6855cdc5.png[/img]
2016 Switzerland - Final Round, 5
Let $ABC$ be a right triangle with $\angle ACB = 90^o$ and M the center of $AB$. Let $G$ br any point on the line $MC$ and $P$ a point on the line $AG$, such that $\angle CPA = \angle BAC$ . Further let $Q$ be a point on the straight line $BG$, such that $\angle BQC = \angle CBA$ . Show that the circles of the triangles $AQG$ and $BPG$ intersect on the segment $AB$.
1984 All Soviet Union Mathematical Olympiad, 378
The circle with the centre $O$ is inscribed in the triangle $ABC$ . The circumference touches its sides $[BC], [CA], [AB]$ in $A_1, B_1, C_1$ points respectively. The $[AO], [BO], [CO]$ segments cross the circumference in $A_2, B_2, C_2$ points respectively. Prove that lines $(A_1A_2),(B_1B_2)$ and $(C_1C_2)$ intersect in one point.
Geometry Mathley 2011-12, 11.4
Let $ABC$ be a triangle and $P$ be a point in the plane of the triangle. The lines $AP,BP, CP$ meets $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Let $A_2,B_2,C_2$ be the Miquel point of the complete quadrilaterals $AB_1PC_1BC$, $BC_1PA_1CA$, $CA_1PB_1AB$. Prove that the circumcircles of the triangles $APA_2$,$BPB_2$, $CPC_2$, $BA_2C$, $AB_2C$, $AC_2B$ have a point of concurrency.
Nguyễn Văn Linh
2011 Sharygin Geometry Olympiad, 3
The line passing through vertex $A$ of triangle $ABC$ and parallel to $BC$ meets the circumcircle of $ABC$ for the second time at point $A_1$. Points $B_1$ and $C_1$ are defined similarly. Prove that the perpendiculars from $A_1, B_1, C_1$ to $BC, CA, AB$ respectively concur.
2020 Greece JBMO TST, 1
Let $ABC$ be a triangle with $AB>AC$. Let $D$ be a point on side $AB$ such that $BD=AC$. Consider the circle $\gamma$ passing through point $D$ and tangent to side $AC$ at point $A$. Consider the circumscribed circle $\omega$ of the triangle $ABC$ that interesects the circle $\gamma$ at points $A$ and $E$. Prove that point $E$ is the intersection point of the perpendicular bisectors of line segments $BC$ and $AD$.
Geometry Mathley 2011-12, 5.3
Let $ABC$ be an acute triangle, not being isoceles. Let $\ell_a$ be the line passing through the points of tangency of the escribed circles in the angle $A$ with the lines $AB, AC$ produced. Let $d_a$ be the line through $A$ parallel to the line that joins the incenter $I$ of the triangle $ABC$ and the midpoint of $BC$. Lines $\ell_b, d_b, \ell_c, d_c$ are defined in the same manner. Three lines $\ell_a, \ell_b, \ell_c$ intersect each other and these intersections make a triangle called $MNP$. Prove that the lines $d_a, d_b$ and $d_c$ are concurrent and their point of concurrency lies on the Euler line of the triangle $MNP$.
Lê Phúc Lữ
2009 Korea Junior Math Olympiad, 2
In an acute triangle $\triangle ABC$, let $A',B',C'$ be the reflection of $A,B,C$ with respect to $BC,CA,AB$. Let $D = B'C \cap BC'$, $E = CA' \cap C'A$, $F = A'B \cap AB'$. Prove that $AD,BE,CF$ are concurrent
2013 Switzerland - Final Round, 10
Let $ABCD$ be a tangential quadrilateral with $BC> BA$. The point $P$ is on the segment $BC$, such that $BP = BA$ . Show that the bisector of $\angle BCD$, the perpendicular on line $BC$ through $P$ and the perpendicular on $BD$ through $A$, intersect at one point.
Geometry Mathley 2011-12, 13.3
Let $ABCD$ be a quadrilateral inscribed in circle $(O)$. Let $M,N$ be the midpoints of $AD,BC$. A line through the intersection $P$ of the two diagonals $AC,BD$ meets $AD,BC$ at $S, T$ respectively. Let $BS$ meet $AT$ at $Q$. Prove that three lines $AD,BC,PQ$ are concurrent if and only if $M, S, T,N$ are on the same circle.
Đỗ Thanh Sơn
1973 All Soviet Union Mathematical Olympiad, 182
Three similar acute-angled triangles $AC_1B, BA_1C$ and $CB_1A$ are constructed on the outer side of the acute-angled triangle $ABC$. (Equal triples of the angles are $AB_1C, ABC_1, A_1BC$ and $BA_1C, BAC_1, B_1AC$.)
a) Prove that the circles circumscribed around the outer triangles intersect in one point.
b) Prove that the straight lines $AA_1, BB_1$ and $CC_1$ intersect in the same point
2021 Sharygin Geometry Olympiad, 8.2
Three parallel lines $\ell_a, \ell_b, \ell_c$ pass through the vertices of triangle $ABC$. A line $a$ is the reflection of altitude $AH_a$ about $\ell_a$. Lines $b, c$ are defined similarly. Prove that $a, b, c$ are concurrent.
Ukrainian TYM Qualifying - geometry, 2013.15
Inside the acute-angled triangle $ABC$, mark the point $O$ so that $\angle AOB=90^o$, a point $M$ on the side $BC$ such that $\angle COM=90^o$, and a point $N$ on the segment $BO$ such that $\angle OMN = 90^o$. Let $P$ be the point of intersection of the lines $AM$ and $CN$, and let $Q$ be a point on the side $AB$ that such $\angle POQ = 90^o$. Prove that the lines $AN, CO$ and $MQ$ intersect at one point.
2019 Balkan MO Shortlist, G8
Given an acute triangle $ABC$, $(c)$ its circumcircle with center $O$ and $H$ the orthocenter of the triangle $ABC$. The line $AO$ intersects $(c)$ at the point $D$. Let $D_1, D_2$ and $H_2, H_3$ be the symmetrical points of the points $D$ and $H$ with respect to the lines $AB, AC$ respectively. Let $(c_1)$ be the circumcircle of the triangle $AD_1D_2$. Suppose that the line $AH$ intersects again $(c_1)$ at the point $U$, the line $H_2H_3$ intersects the segment $D_1D_2$ at the point $K_1$ and the line $DH_3$ intersects the segment $UD_2$ at the point $L_1$. Prove that one of the intersection points of the circumcircles of the triangles $D_1K_1H_2$ and $UDL_1$ lies on the line $K_1L_1$.
Swiss NMO - geometry, 2008.8
Let $ABCDEF$ be a convex hexagon inscribed in a circle . Prove that the diagonals $AD, BE$ and $CF$ intersect at one point if and only if $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$