Found problems: 257
We´re given an inscriptible quadrilateral $ DEFG $ having some vertices on the sides of a triangle $ ABC, $ and some vertices (at least one of them) coinciding with the vertices of the same triangle. Knowing that the lines $ DF $ and $ EG $ aren´t parallel, find the locus of their intersection.
[i]Dan Brânzei[/i]
Circles $\Gamma_1$ and $\Gamma_2$ meet at points $X$ and $Y$. A circle $S_1$ touches internally $\Gamma_1$ at $A$ and $\Gamma_2$ externally at $B$. A circle $S_2$ touches $\Gamma_2$ internally at $C$ and $\Gamma_1$ externally at $D$. Prove that the points $A, B, C, D$ are either collinear or concyclic.
(A. Voidelevich)
Let $ABC$ be a triangle. $M$ the midpoint of $AB$ and $L$ the midpoint of $BC$. We denote by $G$ the intersection of $AL$ with $CM$ and we take $E$ a point such that $G$ is the midpoint of the segment $AE$. Prove that the quadrilateral $MCEB$ is cyclic if and only if $MB = BG$.
Let $ABC$ be an acute triangle orthocenter angle $H$. Let $\omega_1$ be the circle tangent to $BC$ at $B$ and passing through $H$ and $\omega_2$ the circle tangent to $BC$ at $C$ and passing through through $H$. A line $\ell$ passing through $H$ intersects the circles $\omega_1$ and $\omega_2$ at points $D$ and $E$, respectively (with $D$ and $E$ other than $H$). Lines $BD$ and $CE$ intersect at $F$, the lines $\ell$ and $AF$ intersect at $X$ and the circles $\omega_1$ and $\omega_2$ intersect at the points $P$ and $H$. Prove that the points $A, H, P$ and $X$ are still on the same circle.
Given an acute-angled $\vartriangle ABC$ with orthocenter $H$ and altitude $CC_1$. Points $D, E$ and $F$ lie on the segments $AC$, $BC$ and $AB$ respectively, so that $DE \parallel AB$ and $EF \parallel AC$. Denote by $Q$ the symmetric point of $H$ wrt to the midpoint of $DE$. Let $BD \cap CF = P$. If $HP \parallel AB$, prove that the points $C_1, D, Q$ and $E$ lie on a circle.
Let $\triangle ABC$ be a triangle and let $X$ be a point in the interior of the triangle. The second intersection points of the lines $XA,XB$ and $XC$ with the circumcircle of $\triangle ABC$ are $P,Q$ and $R$. Let $U$ be a point on the ray $XP$ (these are the points on the line $XP$ such that $P$ and $U$ lie on the same side of $X$). The line through $U$ parallel to $AB$ intersects $BQ$ in $V$ . The line through $U$ parallel to $AC$ intersects $CR$ in $W$. Prove that $Q, R, V$ , and $W$ lie on a circle.
Let $A, B, C$ and $D$ be a triharmonic quadruple of points, i.e $AB\cdot CD = AC \cdot BD = AD \cdot BC.$ Let $A_1$ be a point distinct from $A$ such that the quadruple $A_1, B, C$ and $D$ is triharmonic. Points $B_1, C_1$ and $D_1$ are defined similarly. Prove that
a) $A, B, C_1, D_1$ are concyclic;
b) the quadruple $A_1, B_1, C_1, D_1$ is triharmonic.
Let $ABC$ be a triangle, $D$ the midpoint of side $BC$ and $E$ the intersection point of the bisector of angle $\angle BAC$ with side $BC$. The perpendicular bisector of $AE$ intersects the bisectors of angles $\angle CBA$ and $\angle CDA$ at $M$ and $N$, respectively. The bisectors of angles $\angle CBA$ and $\angle CDA$ intersect at $P$ . Prove that points $A, M, N, P$ are concyclic.
Let $ABC$ be a scalene triangle with incenter $I$ and let $AI$ meet the circumcircle of triangle $ABC$ again at $M$. The incircle $\omega$ of triangle $ABC$ is tangent to sides $AB, AC$ at $D, E$, respectively. Let $O$ be the circumcenter of triangle $BDE$ and let $L$ be the intersection of $\omega$ and the altitude from $A$ to $BC$ so that $A$ and $L$ lie on the same side with respect to $DE$. Denote by $\Omega$ a circle centered at $O$ and passing through $L$, and let $AL$ meet $\Omega$ again at $N$.
Prove that the lines $LD$ and $MB$ meet on the circumcircle of triangle $LNE$.
Let $ABCD$ be a convex cyclic quadrilateral with diagonals $AC$ and $BD$. Each of the four vertixes are reflected across the diagonal on which the do not lie.
(a) Investigate when the four points thus obtained lie on a straight line and give as simple an equivalent condition as possible to the cyclic quadrilateral $ABCD$ for it.
(b) Show that in all other cases the four points thus obtained lie on one circle.
(Theresia Eisenkölbl)
In the plane there are six different points $A, B, C, D, E, F$ such that $ABCD$ and $CDEF$ are parallelograms. What is the maximum number of those points that can be located on one circle?
In a circle $O$, there are six points, $A,B,C,D,E, F$ in a counterclockwise order. $BD \perp CF$, and $CF,BE,AD$ are concurrent. Let the perpendicular from $B$ to $AC$ be $M$, and the perpendicular from $D$ to $CE$ be $N$. Prove that $AE // MN$.
Given is a pyramid $SABCD$ whose base is a convex quadrilateral $ ABCD $ with perpendicular diagonals $ AC $ and $ BD $, and the orthogonal projection of vertex $S$ onto the base is the point $0$ of the intersection of the diagonals of the base. Prove that the orthogonal projections of point $O$ onto the lateral faces of the pyramid lie on the circle.
Given a triangle $PQR$, the inscribed circle $\omega$ which touches the sides $QR, RP$ and $PQ$ at points $A, B$ and $C$, respectively, and $AB^2 + AC^2 = 2BC^2$. Prove that the point of intersection of the segments $PA, QB$ and $RC$, the center of the circle $\omega$, the point of intersection of the medians of the triangle $ABC$, the point $A$ and the midpoints of the segments $AC$ and $AB$ lie on one circle.
In triangle $ABC$, on side $AC$ there are points $D$ and $E$, that $AB = AD$ and $BE = EC$ ($E$ between $A$ and $D$). Point $F$ is midpoint of arc $BC$ of circumcircle of triangle $ABC$. Prove that the points $B, E, D, F$ lie on the same circle.
Given a quadrilateral $ABCD$, inscribed in a circle $\omega$ such that $AB=AD$ and $CB=CD$ . Take the point $P \in \omega$. Let the vertices of the quadrilateral $Q_1Q_2Q_3Q_4$ be symmetric to the point P wrt the lines $AB$, $BC$, $CD$, and $DA$, respectively.
a) Prove that the points symmetric to the point $P$ wrt lines $Q_1Q_22, Q_2Q_3, Q_3Q_4$ and $Q_4Q_1$, lie on one line.
b) Prove that when the point $P$ moves in a circle $\omega$, then all such lines pass through one common point.
A point $F$ inside a triangle $ABC$ is chosen so that $\angle AFB = \angle BFC = \angle CFA$. The line passing through $F$ and perpendicular to $BC$ meets the median from $A$ at point $A_1$. Points $B_1$ and $C_1$ are defined similarly. Prove that the points $A_1, B_1$, and $C_1$ are three vertices of some regular hexagon, and that the three remaining vertices of that hexagon lie on the sidelines of $ABC$.
Point $M$ on side $AB$ of quadrilateral $ABCD$ is such that quadrilaterals $AMCD$ and $BMDC$ are circumscribed around circles centered at $O_1$ and $O_2$ respectively. Line $O_1O_2$ cuts an isosceles triangle with vertex $M$ from angle $CMD$. prove that $ABCD$ is a cyclc quadrilateral.
Given an acute triangle $ABC, H$ is the foot of the altitude drawn from the point $A$ on the line $BC, P$ and $K \ne H$ are arbitrary points on the segments $AH$ and$ BC$ respectively. Segments $AC$ and $BP$ intersect at point $B_1$, lines $AB$ and $CP$ at point $C_1$. Let $X$ and $Y$ be the projections of point $H$ on the lines $KB_1$ and $KC_1$, respectively. Prove that points $A, P, X$ and $Y$ lie on one circle.
The circles $\gamma$ and $\delta$ are internally tangent to the circle $\omega$ at $A$ and $B$. From $A$, draw two tangent lines $\ell_1, \ell_2$ to $\delta$, . From $B$ draw two tangent lines $t_1, t_2$ to $\gamma$ . Let $\ell_1$ intersect $t_1$ at $X$ and $\ell_2$ intersect $t_2$ at $Y$ . Prove that the quadrilateral $AX BY$ is cyclic.
Nguyen Van Linh, High School of Natural Sciences, Hanoi National University
Let the points $M$ and $N$ be the intersections of the inscribed circle of the right-angled triangle $ABC$, with sides $AB$ and $CA$ respectively , and points $P$ and $Q$ respectively be the intersections of the ex-scribed circles opposite to vertices $B$ and $C$ with direction $BC$. Prove that the quadrilateral $MNPQ$ is a cyclic if and only if the triangle $ABC$ is right-angled with a right angle at the vertex $A$.
Let $ABC$ be an acute-angled triangle. Point $P$ is such that $AP = AB$ and $PB\parallel AC$. Point $Q$ is such that $AQ = AC$ and $CQ\parallel AB$. Segments $CP$ and $BQ$ meet at point $X$. Prove that the circumcenter of triangle $ABC$ lies on the circle $(PXQ)$.
In the triangle $ABC$ the edge $BC$ has length $a$, the edge $AC$ length $b$, and the edge $AB$ length $c$. Extend all the edges at both ends – by the length $a$ from the vertex $A, b$ from $B$, and $c$ from $C$. Show that the six endpoints of the extended edges all lie on a common circle.
[img]https://cdn.artofproblemsolving.com/attachments/8/7/14c8c6a4090d4fade28893729a510d263e7abb.png[/img]
Consider points $C_1, C_2$ on the side $AB$ of a triangle $ABC$, points $A_1, A_2$ on the side $BC$ and points $B_1 , B_2$ on the side $CA$ such that these points divide the corresponding sides to three equal parts. It is known that all the points $A_1, A_2, B_1, B_2 , C_1$ and $C_2$ are concyclic. Prove that triangle $ABC$ is equilateral.
$ABC$ 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 $k$.