Found problems: 257
Four cones are given with a common vertex and the same generatrix, but with, generally speaking, different radii of the bases. Each of them is tangent to two others. Prove that the four tangent points of the circles of the bases of the cones lie on the same circle.
Let the triangle $ABC$ be with $| AB | > | AC |$. Point M is the midpoint of the side $[BC]$, and point $I$ is the center of the circle inscribed in the triangle ABC such that the relation $| AI | = | MI |$. Prove that points $A, B, M, I$ are located on the same circle.
Given a scalene acute triangle $ABC$, let M be the midpoints of its side $BC$ and $N$ the midpoint of the arc $BAC$ of its circumcircle. Let $\omega$ be the circle with diameter $BC$ and $D$, $E$ its intersections with the bisector of angle $\angle BAC$. Points $D'$, $E'$ lie on $\omega$ such that $DED'E' $ is a rectangle. Prove that $D'$, $E'$, $M$, $N$ lie on a single circle.
[i] (Patrik Bak)[/i]
Prove that the diagonals of a quadrilateral are perpendicular to each other if and only if the midpoints of it's sides lie on a circle.
Let $ABC$ be an acute angled triangle. Let the altitude from $C$ to $AB$ meet $AB$ at $C'$ and have midpoint $M$, and let the altitude from $B$ to $AC$ meet $AC$ at $B'$ and have midpoint $N$. Let $P$ be the point of intersection of $AM$ and $BB'$ and $Q$ the point of intersection of $AN$ and $CC'$. Prove that the point $M, N, P$ and $Q$ lie on a circle.
Let $ABCD$ be a quadrilateral. The circumcircle of the triangle $ABC$ intersects the sides $CD$ and $DA$ in the points $P$ and $Q$ respectively, while the circumcircle of $CDA$ intersects the sides $AB$ and $BC$ in the points $R$ and $S$. The lines $BP$ and $BQ$ intersect the line $RS$ in the points $M$ and $N$ respectively. Prove that the points $M, N, P$ and $Q$ lie on the same circle.
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 center of the circumcircle of the acute triangle $ABC$ is $O$. The line $AO$ intersects $BC$ at $D$. On the sides $AB$ and $AC$ of the triangle, choose points $E$ and $F$, respectively, so that the points $A, E, D, F$ lie on the same circle. Let $E'$ and $F'$ projections of points $E$ and $F$ on side $BC$ respectively. Prove that length of the segment $E'F'$ does not depend on the position of points $E$ and $F$.
The median $CM$ is drawn in the triangle $ABC$ intersecting bisector angle $BL$ at point $O$. Ray $AO$ intersects side $BC$ at point $K$, beyond point $K$ draw the segment $KT = KC$. On the ray $BC$ beyond point $C$ draw a segment $CN = BK$. Prove that is a quadrilateral $ABTN$ is cyclic if and only if $AB = AK$.
(Vladislav Yurashev)
Given a circle centered at point $O$, with $AB$ as the diameter. Point $C$ lies on the extension of line $AB$ so that $B$ lies between $A$ and $C$, and the line through $C$ intersects the circle at points $D$ and $E$ (where $D$ lies between $C$ and $E$). $OF$ is the diameter of the circumcircle of triangle $OBD$, and the extension of the line $CF$ intersects the circumcircle of triangle $OBD$ at point $G$. Prove that the points $O, A, E, G$ lie on a circle.
Circles $S_1$ and $S_2$ intersect at points $M$ and $N$. Through point $A$ of circle $S_1$, draw straight lines $AM$ and $AN$ intersecting $S_2$ at points $B$ and $C$, and through point $D$ of circle $S_2$, draw straight lines $DM$ and $DN$ intersecting $S_1$ at points $E$ and $F$, and $A$, $E$, $F$ lie along one side of line $MN$, and $D$, $B$, $C$ lie on the other side (see figure). Prove that if $AB = DE$, then points $A$, $F$, $C$ and $D$ lie on the same circle, the position of the center of which does not depend on choosing points $A$ and $D$.
[img]https://cdn.artofproblemsolving.com/attachments/7/0/d1f9c2f39352e2b39e55bd2538677073618ef9.png[/img]
Let $k$ be a circle with diameter $AB$ and centre $O$. Let C be an arbitrary point on the circle different from $A$ and $B$. Let $D$ be the point for which $O$, $B$, $D$ and $C$ (in this order) are the four vertices of a parallelogram. Let $E$ be the intersection of the line $BD$ and the circle $k$, and let $F$ be the orthocenter of the triangle $OAC$. Prove that the points $O, D, E, C, F$ lie on a circle.
Let $\triangle ABC$ be a triangle. Let $O$ be the circumcenter of triangle $\triangle ABC$ and $P$ a variable point in line segment $BC$. The circle with center $P$ and radius $PA$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $R$ and $RP$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $Q$. Show that points $A$, $O$, $P$ and $Q$ are concyclic.
The points $A, B, C, D, P$ lie on an circle as shown in the figure such that $\angle AP B = \angle BPC = \angle CPD$. Prove that the lengths of the segments are denoted by $a, b, c, d$ by $\frac{a + c}{b + d} =\frac{b}{c}$.
[img]https://cdn.artofproblemsolving.com/attachments/a/2/ba8965f5d7d180426db26e8f7dd5c7ad02c440.png[/img]
There are four points $A, B, C, D$ on a circle. Circles are drawn through each pair of neighboring points. Denote the intersection points of neighboring circles by $A_1, B_1, C_1, D_1$. (Some of these points may coincide with previously given ones.) Prove that points $A_1, B_1, C_1, D_1$ lie on one circle.
Let $O$ be the centre of the circle touching the side $AC$ of triangle $ABC$ and the continuations of the sides $BA$ and $BC$. $D$ is the centre of the circle passing through the points $A$, $B$ and $O$. Prove that the points $A$, $B$, $C$ and $D$ lie on a circle.
(YF Akurlich)
Let $ABC$ be an acute triangle such that $AB>BC>AC$. Let $D$ be a point different from $C$ on the segment $BC$, such that $AC=AD$. Let $H$ be the orthocenter of triangle $ABC$ and let $A_1$ and $B_1$ be the intersections of the heights from $A$ and $B$ to the opposite sides, respectively. Let $E$ be the intersection of the lines $A_1B_1$ and $DH$. Prove that $B$, $D$, $B_1$, $E$ are concyclic.
Let $G$ be the centroid of $\triangle ABC $ and let $D, E $ and $F$ be the midpoints of the line segments $BC, CA $ and $AB$ respectively. Suppose the circumcircle of $\triangle ABC $ meets $AD $ again at $X$, the circumcircle of $\triangle DEF $ meets $BE$ again at $Y$ and the circumcircle of $\triangle DEF $ meets $CF$ again at $Z$. Show that $G, X, Y $ and $Z$ are concyclic.
Given a pointed triangle $ABC$ with orthocenter $H$. Let $D$ be the point such that the quadrilateral $AHCD$ is parallelogram. Let $p$ be the perpendicular to the direction $AB$ through the midpoint $A_1$ of the side $BC$. Denote the intersection of the lines $p$ and $AB$ with $E$, and the midpoint of the length $A_1E$ with $F$. The point where the parallel to the line $BD$ through point $A$ intersects $p$ denote by $G$. Prove that the quadrilateral $AFA_1C$ is cyclic if and only if the lines $BF$ passes through the midpoint of the length $CG$.
Let $P$ be a point in parallelogram $ABCD$ such that $$PA \cdot PC + PB \cdot PD = AB \cdot BC.$$ Prove that the reflections of $P$ over lines $AB$, $BC$, $CD$, and $DA$ are concyclic.
Let $ABC$ be an acute, nonisosceles triangle with inscribe in a circle $(O)$ and has orthocenter $H$. Denote $M,N,P$ as the midpoints of sides $BC,CA,AB$ and $D,E,F$ as the feet of the altitudes from vertices $A,B,C$ of triangle $ABC$. Let $K$ as the reflection of $H$ through $BC$. Two lines $DE,MP$ meet at $X$; two lines $DF,MN$ meet at $Y$.
a) The line $XY$ cut the minor arc $BC$ of $(O)$ at $Z$. Prove that $K,Z,E,F$ are concyclic.
b) Two lines $KE,KF$ cuts $(O)$ second time at $S,T$. Prove that $BS,CT,XY$ are concurrent.
A pyramid with a quadrangular base is given such that each pair of circles inscribed in adjacent faces has a common point. Prove that the touchpoints of these circles with the base of the pyramid lie on one circle.
Let $\vartriangle ABC$ be an acute triangle, with $\angle A> \angle B \ge \angle C$. Let $D, E$ and $F$ be the tangency points between the incircle of triangle and sides $BC, CA, AB$, respectively. Let $J$ be a point on $(BD)$, $K$ a point on $(DC)$, $L$ a point on $(EC)$ and $M$ a point on $(FB)$, such that $$AF = FM = JD = DK = LE = EA.$$Let $P$ be the intersection point between $AJ$ and $KM$ and let $Q$ be the intersection point between $AK$ and $JL$. Prove that $PJKQ$ is cyclic.
Let $ ABC $ be an inscribed triangle in circle $ (O) $. Let $ D $ be the intersection of the two tangent lines of $ (O) $ at $ B $ and $ C $. The circle passing through $ A $ and tangent to $ BC $ at $ B $ intersects the median passing $ A $ of the triangle $ ABC $ at $ G $. Lines $ BG, CG $ intersect $ CD, BD $ at $ E, F $ respectively.
a) The line passing through the midpoint of $ BE $ and $ CF $ cuts $ BF, CE $ at $ M, N $ respectively. Prove that the points $ A, D, M, N $ belong to the same circle.
b) Let $ AD, AG $ intersect the circumcircle of the triangles $ DBC, GBC $ at $ H, K $ respectively. The perpendicular bisectors of $ HK, HE$, and $HF $ cut $ BC, CA$, and $AB $ at $ R, P$, and $Q $ respectively. Prove that the points $ R, P$, and $Q $ are collinear.
On the $ AB $ side of the triangle $ ABC $, points $ X $ and $ Y $ are chosen, on the side of $ AC $ is a point of $ Z $, and on the side of $ BC $ is a point of $ T $. Wherein $ XZ \parallel BC $, $ YT \parallel AC $. Line $ TZ $ intersects the circumscribed circle of triangle $ ABC $ at points $ D $ and $ E $. Prove that points $ X $, $ Y $, $ D $ and $ E $ lie on the same circle.