Found problems: 478
Let $ABC$ be a triangle. On the extensions of sides $AB$ and $CB$ towards $B$, points $C_1$ and $A_1$ are taken, respectively, so that $AC = A_1C = AC_1$. Prove that circumscribed circles of triangles $ABA_1$ and $CBC_1$ intersect on the bisector of angle $B$.
Point $O$ is the circumcenter of acute-angled triangle $ABC$, points $A_1,B_1, C_1$ are the bases of its altitudes. Points $A', B', C'$ lying on lines $OA_1, OB_1, OC_1$ respectively are such that quadrilaterals $AOBC', BOCA', COAB'$ are cyclic. Prove that the circumcircles of triangles $AA_1A', BB_1B', CC_1C'$ have a common point.
Given an acute triangle $ABC$ with altitudes $AD$ and $BE$ intersecting at $H$, $M$ is the midpoint of $AB$. A nine-point circle of $ABC$ intersects with a circumcircle of $ABH$ on $P$ and $Q$ where $P$ lays on the same side of $A$ (with respect to $CH$). Prove that $ED, PH, MQ$ are concurrent on circumcircle $ABC$
Let $ABC$ be a non-isosceles triangle. The incircle $(I)$ of the triangle touches sides $BC,CA,AB$ at $A_0,B_0$, and $C_0$. Points $A_1,B_1$, and $C_1$ are on $BC,CA,AB$ such that $BA1 = CA_0, CB_1 = AB_0, AC_1 = BC_0$. Prove that the circumcircles $(IAA1), (IBB_1), (ICC_1)$ pass all through a common point, distinct from $I$.
Nguyễn Minh Hà
Let $\omega$ be the circumscribed circle of the triangle $ABC$, in which $AC< AB$, $K$ is the center of the arc $BAC$, $KW$ is the diameter of the circle $\omega$. The circle $\gamma$ is inscribed in the curvilinear triangle formed by the segments $BC$, $AB$ and the arc $AC$ of the circle $\omega$. It turned out that circle $\gamma$ also touches $KW$ at point $F$. Let $I$ be the center of the triangle $ABC$, $M$ is the midpoint of the smaller arc $AK$, and $T$ is the second intersection point of $MI$ with the circle $\omega$. Prove that lines $FI$, $TW$ and $BC$ intersect at one point.
(Mykhailo Sydorenko)
Let $BE{}$ and $CF$ be heights in the acute-angled triangle $ABC{}$ and let $O{}$ be its circumcenter. The points $M{}$ and $N{}$ are selected on the side $BC{}$ so that $BM=CN.{}$ The line $BE{}$ intersects the circle $(MBF)$ a second time at $P{}$ and the line $CF{}$ intersects the circle $(NCE)$ a second time at $Q.{}$ Prove that the lines $PF, QE$ and $AO{}$ intersect at the same point.
[i]Proposed by Luu Dong[/i]
The circle k intersects the sides $BC$, $CA$, $AB$ of triangle $ABC$ in points $A_1$, $A_2$, $B_1$, $B_2$, $C_1$, $C_2$. The perpendiculars to $BC$, $CA$, $AB$ through $A_1$, $B_1$, $C_1$, respectively, meet at a point $M$. Prove that the three perpendiculars to $BC$, $CA$, $AB$ through $A_2$, $B_2$, and $C_2$, respectively, also meet in one point.
A circle $c$ with center $A$ passes through the vertices $B$ and $E$ of a regular pentagon $ABCDE$ . The line $BC$ intersects the circle $c$ for second time at point $F$. The point $G$ on the circle $c$ is chosen such that $| F B | = | FG |$ and $B \ne G$. Prove that the lines $AB, EF$ and $DG$ intersect at one point.
Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.
Four points are given $A, B, C, D$. Points $A_1, B_1, C_1,D_1$ are orthocenters of the triangles $BCD, CDA, DAB, ABC$ and $A_2, B_2, C_2,D_2$ are orthocenters of the triangles $B_1C_1D_1, C_1D_1A_1, D_1A_1B_1,A_1B_1C_1$ etc. Prove that the circles passing through the midpoints of the sides of all the triangles intersect at one point.
Let $ABC$ be an acute triangle. Squares $AA_1A_2A_3$, $BB_1B_2B_3$ and $CC_1C_2C_3$ are located such that the lines $A_1A_2$, $B_1B_2$, $C_1C_2$ pass through the points $B$, $C$ and $A$ respectively and the lines $A_2A_3$, $B_2B_3$, $C_2C_3$ pass through the points $C$, $A$ and $B$ respectively. Prove that
(a) the lines $AA_2$, $B_1B_2$ and $C_1C_3$ intersect at one point.
(b) the lines $AA_2$, $BB_2$ and $CC_2$ intersect at one point.
(Mykhailo Plotnikov)
[img]https://cdn.artofproblemsolving.com/attachments/3/d/ad2fe12ae2c82d04b48f5e683b7d54e0764baf.png[/img]
Let $ABCD$ be a parallelogram with vertices in order clockwise and let $E$ be the intersection of its diagonals. The circle of diameter $DE$ intersects the segment $AD$ at $L$ and $EC$ at $H$. The circumscribed circle of $LEB$ intersects the segment $BC$ at $O$. Prove that the lines $HD$ , $LE$ and $BC$ are concurrent if and only if $EO = EC$.
Two triangles $ABC$ and $PQR$ have the same circumcircles. Let $E_a, E_b, E_c$ be the centers of the Euler circles of triangles $PBC, QCA, RAB$. Assume that $d_a$ is a line through $Ea$ parallel to $AP$, $d_b, d_c$ are defined in the same manner. Prove that three lines $d_a, d_b, d_c$ are concurrent.
Nguyễn Tiến Lâm, Trần Quang Hùng
Let $ABC$ denote a triangle with $AC\ne BC$. Let $I$ and $U$ denote the incenter and circumcenter of the triangle $ABC$, respectively. The incircle touches $BC$ and $AC$ in the points $D$ and E, respectively. The circumcircles of the triangles $ABC$ and $CDE$ intersect in the two points $C$ and $P$. Prove that the common point $S$ of the lines $CU$ and $P I$ lies on the circumcircle of the triangle $ABC$.
[i](Karl Czakler)[/i]
Let $ABCD$ be a rhombus with $\angle BAD < 90^o$. The circle passing through $D$ with center $A$ intersects the line $CD$ a second time in point $E$. Let $S$ be the intersection of the lines $BE$ and $AC$. Prove that the points $A$, $S$, $D$ and $E$ lie on a circle.
[i](Karl Czakler)[/i]
Let $ABCDEF$ be a convex hexagon. In the hexagon there is a point $K$, such that $ABCK,DEFK$ are both parallelograms. Prove that the three lines connecting $A,B,C$ to the midpoints of segments $CE,DF,EA$ meet at one point.
Let $ABC$ be a scalene triangle, and $A_o$, $B_o,$ $C_o$ be the midpoints of $BC$, $CA$, $AB$ respectively. The bisector of angle $C$ meets $A_oCo$ and $B_oC_o$ at points $B_1$ and $A_1$ respectively. Prove that the lines $AB_1$, $BA_1$ and $A_oB_o$ concur.
The bisectors $AA_1$ and $CC_1$ of triangle $ABC$ intersect at point $I$. The circumscribed circles of triangles $AIC_1$ and $CIA_1$ intersect the arcs $AC$ and $BC$ (not containing points $B$ and $A$ respectively) of the circumscribed circle of triangle $ABC$ at points $C_2$ and $A_2$, respectively. Prove that lines $A_1A_2$ and $C_1C_2$ intersect on the circumscribed circle of triangle $ABC$.
Let $ABCDEF$ be a convex hexagon with $AB = BC, CD = DE$ and $EF = FA$.
Prove that the lines through $C,E,A$ perpendicular to $BD,DF,FB$ are concurrent.
Given a line segment $AB$ and a point $C$ lies inside it such that $AB=3 \cdot AC$ . Construct a parallelogram $ACDE$ such that $AC=DE=CE>AR$. Let $Z$ be a point on $AC$ such that $\angle AEZ=\angle ACE =\omega$. Prove that the line passing through point $B$ and perpendicular on side $EC$, and the line passing through point $D$ and perpendicular on side $AB$, intersect on point , let it be $K$, lying on line $EZ$.
Let the inscribed circle $ \omega $ of triangle $ ABC $ touch the side $ BC $ at the point $ A '$. Let $ AA '$ intersect $ \omega $ at $ P \neq A $. Let $ CP $ and $ BP $ intersect $ \omega $, respectively, at points $ N $ and $ M $ other than $ P $. Prove that $ AA ', BN $ and $ CM $ intersect at one point.
Prove that all the heights of a tetrahedron intersect at one point if and only if the sums of the squares of the opposite edges are equal.
A non-isosceles trapezoid $ABCD$ ($AB // CD$) is given. An arbitrary circle passing through points $A$ and $B$ intersects the sides of the trapezoid at points $P$ and $Q$, and the intersect the diagonals at points $M$ and $N$. Prove that the lines $PQ, MN$ and $CD$ are concurrent.
Let $ABC$ be a triangle with $AB < AC < BC$. On the side $BC$ we consider points $D$
and $E$ such that $BA = BD$ and $CE = CA$. Let $K$ be the circumcenter of triangle $ADE$ and
let $F$, $G$ be the points of intersection of the lines $AD$, $KC$ and $AE$, $KB$ respectively. Let $\omega_1$ be
the circumcircle of triangle $KDE$, $\omega_2$ the circle with center $F$ and radius $FE$, and $\omega_3$ the circle
with center $G$ and radius $GD$.
Prove that $\omega_1$, $\omega_2$, and $\omega_3$ pass through the same point and that this point of intersection lies on the line $AK$.
A point $P$ is given inside a triangle $ABC$. Let $D,E,F$ be the midpoints of $AP,BP,CP$, and let $L,M,N$ be the intersection points of $ BF$ and $CE, AF$ and $CD, AE$ and $BD$, respectively.
(a) Prove that the area of hexagon $DNELFM$ is equal to one third of the area of triangle $ABC$.
(b) Prove that $DL,EM$, and $FN$ are concurrent.