Found problems: 3585
Let $ABC$ be an acute-angled triangle in which $\angle ABC$ is the largest angle. Let $O$ be its circumcentre. The perpendicular bisectors of $BC$ and $AB$ meet $AC$ at $X$ and $Y$ respectively. The internal angle bisectors of $\angle AXB$ and $\angle BYC$ meet $AB$ and $BC$ at $D$ and $E$ respectively. Prove that $BO$ is perpendicular to $AC$ if $DE$ is parallel to $AC$.
Altitudes $AA_1, BB_1, CC_1$ of acute triangle $ABC$ intersect at point $H$. On the tangent drawn from point $C$ to the circle $(AB_1C_1)$, the perpendicular $HQ$ is drawn (the point $Q$ lies inside the triangle $ABC$). Prove that the circle passing through the point $B_1$ and touching the line $AB$ at point $A$ is also tangent to line $A_1Q$.
The circumcircle of acute $\triangle ABC$ has center $O$. The line passing through point $O$ perpendicular to $\overline{OB}$ intersects lines $AB$ and $BC$ at $P$ and $Q$, respectively. Also $AB=5$, $BC=4$, $BQ=4.5$, and $BP=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $\ell$ be a tangent line to $\Gamma$, and let $\ell_a, \ell_b$ and $\ell_c$ be the lines obtained by reflecting $\ell$ in the lines $BC$, $CA$ and $AB$, respectively. Show that the circumcircle of the triangle determined by the lines $\ell_a, \ell_b$ and $\ell_c$ is tangent to the circle $\Gamma$.
[i]Proposed by Japan[/i]
Let $ABC$ be a triangle with circumcircle $\omega$ and $\ell$ a line without common points with $\omega$. Denote by $P$ the foot of the perpendicular from the center of $\omega$ to $\ell$. The side-lines $BC,CA,AB$ intersect $\ell$ at the points $X,Y,Z$ different from $P$. Prove that the circumcircles of the triangles $AXP$, $BYP$ and $CZP$ have a common point different from $P$ or are mutually tangent at $P$.
[i]Proposed by Cosmin Pohoata, Romania[/i]
Let $ABCD$ be a cyclic quadrilateral, so that $|AB| + |CD| = |BC|$. Show that the intersection of the bisector of $\angle DAB$ and $\angle CDA$ lies on the side $BC$.
For each parabola $y = x^2+ px+q$ intersecting the coordinate axes in three distinct points, consider the circle passing through these points. Prove that all these circles pass through a single point, and find this point.
Let $ABC$ be a triangle with $AB > AC$ with incenter $I{}$. The internal bisector of the angle $BAC$ intersects the $BC$ at the point $D{}$. Let $M{}$ the midpoint of the segment $AD{}$, and let $F{}$ be the second intersection point of $MB$ with the circumcircle of the triangle $BIC$. Prove that $AF$ is perpendicular to $FC$.
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.
Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.
Let $ A, B, D, E, F, C $ be six points lie on a circle (in order) satisfy $ AB=AC $ .
Let $ P=AD \cap BE, R=AF \cap CE, Q=BF \cap CD, S=AD \cap BF, T=AF \cap CD $ .
Let $ K $ be a point lie on $ ST $ satisfy $ \angle QKS=\angle ECA $ .
Prove that $ \frac{SK}{KT}=\frac{PQ}{QR} $
Circles $ \omega_{1}$ and $ \omega_{2}$ intersect at points $ A$ and $ B.$ Points $ C$ and $ D$ are on circles $ \omega_{1}$ and $ \omega_{2},$ respectively,
such that lines $ AC$ and $ AD$ are tangent to circles $ \omega_{2}$ and $ \omega_{1},$ respectively.
Let $ I_{1}$ and $ I_{2}$ be the incenters of triangles $ ABC$ and $ ABD,$ respectively. Segments $ I_{1}I_{2}$ and $ AB$ intersect at $ E$.
Prove that: $ \frac {1}{AE} \equal{} \frac {1}{AC} \plus{} \frac {1}{AD}$
In a nonisosceles triangle $ABC$, point $I$ is its incentre and $\Gamma$ is its circumcircle. Points $E$ and $D$ lie on $\Gamma$ and the circumcircle of triangle $BIC$ respectively such that $AE$ and $ID$ are both perpendicular to $BC$. Let $M$ be the midpoint of $BC$, $N$ be the midpoint of arc $BC$ on $\Gamma$ containing $A$, $F$ is the point of tangency of the $A-$excircle on $BC$, and $G$ is the intersection of line $DE$ with $\Gamma$. Prove that lines $GM$ and $NF$ intersect at a point located on $\Gamma$.
(Possibly proposed by Farras Faddila)
Lei it \(ABCD\) be a non-cyclical, convex quadrilateral, with no parallel sides.
The lines \(AB\) and \(CD\) meet in \(E\).
Let it \(M \not= E\) be the intersection of circumcircles of \(ADE\) and \(BCE\).
The internal angle bisectors of \(ABCD\) form an convex, cyclical quadrilateral with circumcenter \(I\).
The external angle bisectors of \(ABCD\) form an convex, cyclical quadrilateral with circumcenter \(J\).
Show that \(I,J,M\) are colinear.
Points $P$ and $Q$ lie respectively on sides $AB$ and $AC$ of a triangle $ABC$ and $BP=CQ$. Segments $BQ$ and $CP$ cross at $R$. Circumscribed circles of triangles $BPR$ and $CQR$ cross again at point $S$ different from $R$. Prove that point $S$ lies on the bisector of angle $BAC$.
Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.
On the side $AB$ of the cyclic quadrilateral $ABCD$ there is a point $X$ such that diagonal $AC$ bisects the segment $DX$, and the diagonal $BD$ bisects the segment $CX$. What is the smallest possible ratio $|AB | : |CD|$ in such a quadrilateral ?
The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ meet at $D$. The circumcircle of $\triangle BCD$ meets sides $AC$ and $AB$ again at $E$ and $F$ respectively. Let $O$ be the circumcentre of $\triangle ABC$. Show that $AO$ is perpendicular to $EF$.
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.
Let $\omega_1$ and $\omega_2$ with center $O_1$ and $O_2$ respectively, meet at points $A$ and $B$. Let $X$ and $Y$ be points on $\omega_1$. Lines $XA$ and $Y A$ meet $\omega_2$ at $Z$ and $W$, respectively, such that $A$ lies between $X$ and $Z$ and between $Y$ and $W$. Let $M$ be the midpoint of $O_1O_2$, $S$ be the midpoint of $XA$ and $T$ be the midpoint of $W A$. Prove that $MS = MT$ if and only if $X,~ Y ,~ Z$ and $W$ are concyclic.
Let $K, L, M$ and $N$ be the midpoints of the sides $AB, BC, CD$ and $DA$ of a cyclic quadrilateral $ABCD$. Let $P$ be the point of intersection of $AC$ and $BD$. Prove that the circumradii of triangles $PKL, PLM, PMN$ and $PNK$ are equal to one another.
Rectangle $ABCD$ is inscribed in circle $O$. Let the projections of a point $P$ on minor arc $CD$ onto $AB,AC,BD$ be $K,L,M$, respectively. Prove that $\angle LKM=45$if and only if $ABCD$ is a square.
Let $\triangle{ABC}$ be a triangle with orthocentre $H$. The tangent lines from $A$ to the circle with diameter $BC$ touch this circle at $P$ and $Q$. Prove that $H,P$ and $Q$ are collinear.
Let $ABCD$ be a parallelogram. The interior angle bisector of $\angle ADC$ intersects the line $BC$ in $E$, and the perpendicular bisector of the side $AD$ intersects the line $DE$ in $M$. Let $F= AM \cap BC$. Prove that:
a) $DE=AF$;
b) $AD\cdot AB = DE\cdot DM$.
[i]Daniela and Marius Lobaza, Timisoara[/i]
Let $B, C, D, E$ be four pairwise distinct collinear points and let $A$ be a point not on ine $BC$. Now, let the circumcircle of $\triangle ABC$ meet $AD$ and $AE$ respectively again at $F$ and $G$.
Show that $DEFG$ is cyclic if and only if $AB=AC$.