Found problems: 3585
Let $C_1$ and $C_2$ be two concentric circles with $C_1$ inside $C_2$. Let $P_1$ and $P_2$ be two points on $C_1$ that are not diametrically opposite. Extend the segment $P_1P_2$ past $P_2$ until it meets the circle $C_2$ in $Q_2$. The tangent to $C_2$ at $Q_2$ and the tangent to $C_1$ at $P_1$ meet in a point $X$. Draw from X the second tangent to $C_2$ which meets $C_2$ at the point $Q_1$. Show that $P_1X$ bisects angle $Q_1P_1Q_2$.
The feet of the perpendiculars from the intersection point of the diagonals of a convex cyclic quadrilateral to the sides form a quadrilateral $q$. Show that the sum of the lengths of each pair of opposite sides of $q$ is equal.
Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and \[
\angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \] Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.
Consider the triangles $[ABC]$ and $[EDC]$, right at $A$ and $D$, respectively. Show that, if $E$ is the midpoint of $[AC]$, then $AB <BD$.
[img]https://cdn.artofproblemsolving.com/attachments/c/3/75bc1bda1a22bcf00d4fe7680c80a81a9aaa4c.png[/img]
In acute triangle $ABC$, let $D$ and $E$ be points on sides $AB$ and $AC$ respectively. Let segments $BE$ and $DC$ meet at point $H$. Let $M$ and $N$ be the midpoints of segments $BD$ and $CE$ respectively. Show that $H$ is the orthocenter of triangle $AMN$ if and only if $B,C,E,D$ are concyclic and $BE\perp CD$.
Given circle $O$ with radius $R$, the inscribed triangle $ABC$ is an acute scalene triangle, where $AB$ is the largest side. $AH_A, BH_B,CH_C$ are heights on $BC,CA,AB$. Let $D$ be the symmetric point of $H_A$ with respect to $H_BH_C$, $E$ be the symmetric point of $H_B$ with respect to $H_AH_C$. $P$ is the intersection of $AD,BE$, $H$ is the orthocentre of $\triangle ABC$. Prove: $OP\cdot OH$ is fixed, and find this value in terms of $R$.
(Edited)
Given a triangle $ABC$, a point $P$ is chosen on side $BC$. Points $M$ and $N$ lie on sides $AB$ and $AC$, respectively, such that $MP \parallel AC$ and $NP \parallel AB$. Point $P$ is reflected across $MN$ to point $Q$. Show that triangle $QMB$ is similar to triangle $CNQ$.
[i]Brian Hamrick.[/i]
Let $\mathcal{C}$ be the circumference inscribed in the triangle $ABC,$ which is tangent to sides $BC, AC, AB$ at the points $A' , B' , C' ,$ respectively. The distinct points $K$ and $L$ are taken on $\mathcal{C}$ such that $$\angle AKB'+\angle BKA' =\angle ALB'+\angle BLA'=180^{\circ}.$$
Prove that the points $A', B', C'$ are equidistant from the line $KL.$
Let $ABC$ be a triangle with orthocenter $H$ and circumcenter $O$. Let $K$ be the midpoint of $AH$. point $P$ lies on $AC$ such that $\angle BKP=90^{\circ}$. Prove that $OP\parallel BC$.
Let $ABCDE$ be a convex pentagon with $CD= DE$ and $\angle EDC \ne 2 \cdot \angle ADB$.
Suppose that a point $P$ is located in the interior of the pentagon such that $AP =AE$ and $BP= BC$.
Prove that $P$ lies on the diagonal $CE$ if and only if area $(BCD)$ + area $(ADE)$ = area $(ABD)$ + area $(ABP)$.
(Hungary)
There is a convex quadrilateral $ ABCD $ which satisfies $ \angle A=\angle D $.
Let the midpoints of $ AB, AD, CD $ be $ L,M,N $.
Let's say the intersection point of $ AC, BD $ be $ E $ .
Let's say point $ F $ which lies on $ \overrightarrow{ME} $ satisfies $ \overline{ME}\times \overline{MF}=\overline{MA}^{2} $.
Prove that $ \angle LFM=\angle MFN $. :)
Let $ABC$ be a triangle with $AB\neq AC$. Let the excircle $\omega$ opposite $A$ touch $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. Suppose $X$ and $Y$ are points on the segments $AC$ and $AB$, respectively, such that $XY$ and $BC$ are parallel, and let $\Gamma$ be a circle through $X$ and $Y$ which is externally tangent to $\omega$ at $Z$. Prove that the lines $EF$, $DZ$, and $XY$ are concurrent.
Let $ABC$ be a triangle with circumcentre $O$. The points $D,E,F$ lie in the interiors of the sides $BC,CA,AB$ respectively, such that $DE$ is perpendicular to $CO$ and $DF$ is perpendicular to $BO$. (By interior we mean, for example, that the point $D$ lies on the line $BC$ and $D$ is between $B$ and $C$ on that line.)
Let $K$ be the circumcentre of triangle $AFE$. Prove that the lines $DK$ and $BC$ are perpendicular.
[i]Netherlands (Merlijn Staps)[/i]
Let $ABCD$ be a cyclic quadrilateral. Points $E$ and $F$ lie on the sides $AD$ and $CD$ in such a way that $AE = BC$ and $AB = CF$. Let $M$ be the midpoint of $EF$. Prove that $\angle AMC = 90^{\circ}$.
Let $ABC$ be a triangle and $AB<AC<BC$. Let $D,E$ be points on the side $BC$ and the line $AB$, respectively ($A$ is between $B,E$) such that $BD=BE=AC$. The circumcircle of $\Delta BED$ meets the side $AC$ at $P$ and $BP$ meets the circumcircle of $\Delta ABC$ at $Q$. Prove that:
\[ AQ+CQ=BP. \]
$A,\ B,\ C,\ D,\ E$ and $F$ lie (in that order) on the circumference of a circle. The chords $AD,\ BE$ and $CF$ are concurrent. $P,\ Q$ and $R$ are the midpoints of $AD,\ BE$ and $CF$ respectively. Two further chords $AG \parallel BE$ and $AH \parallel CF$ are drawn. Show that $PQR$ is similar to $DGH$.
Let $D$ and $E$ be foots of altitudes from $A$ and $B$ of triangle $ABC$, $F$ be intersection point of angle bisector from $C$ with side $AB$, and $O$, $I$ and $H$ be circumcenter, center of inscribed circle and orthocenter of triangle $ABC$, respectively. If $\frac{CF}{AD}+ \frac{CF}{BE}=2$, prove that $OI = IH$.
From the point $P$ outside a circle $\omega$ with center $O$ draw the tangents $PA$ and $PB$ where $A$ and $B$ belong to $\omega$.In a random point $M$ in the chord $AB$ we draw the perpendicular to $OM$, which intersects $PA$ and $PB$ in $C$ and $D$. Prove that $M$ is the midpoint $CD$.
Let $ABCD$ be a cyclic quadrilateral, for which $B$ and $C$ are acute angles. $M$ and $N$ are the projections of the vertex $B$ on the lines $AC$ and $AD$, respectively, $P$ and $T$ are the projections of the vertex $D$ on the lines $AB$ and $AC$ respectively, $Q$ and $S$ are the intersections of the pairs of lines $MN$ and $CD$, and $PT$ and $BC$, respectively. Prove the following statements:
a) $NS \parallel PQ \parallel AC$;
b) $NP=SQ$;
c) $NPQS$ is a rectangle if, and only if, $AC$ is a diamteter of the circumscribed circle of quadrilateral $ABCD$.
The diagonals of a cyclic quadrilateral $ABCD$ meet in a point $N.$ The circumcircles of triangles $ANB$ and $CND$ intersect the sidelines $BC$ and $AD$ for the second time in points $A_1,B_1,C_1,D_1.$ Prove that the quadrilateral $A_1B_1C_1D_1$ is inscribed in a circle centered at $N.$
Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$.
[i]Proposed by Hojoo Lee, Korea[/i]
Triangle $ABC$ is an isosceles right triangle with $AB=AC=3$. Let $M$ be the midpoint of hypotenuse $\overline{BC}$. Points $I$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $AI>AE$ and $AIME$ is a cyclic quadrilateral. Given that triangle $EMI$ has area $2$, the length $CI$ can be written as $\frac{a-\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is the value of $a+b+c$?
$
\textbf{(A) }9 \qquad
\textbf{(B) }10 \qquad
\textbf{(C) }11 \qquad
\textbf{(D) }12 \qquad
\textbf{(E) }13 \qquad
$
Let $ABC$ be an acute triangle.A circle with diameter $BC$ meets $AB$ and $AC$ at $E$ and $F$,respectively. $M$ is midpoint of $BC$ and $P$ is point of intersection $AM$ with $EF$. $X$ is an arbitary point on arc $EF$ and $Y$ is the second intersection of $XP$ with a circle with diameter $BC$.Prove that $ \measuredangle XAY=\measuredangle XYM $.
Author:Ali zo'alam , Iran
Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.
In a triangle $ABC$, the angle bisector at $A,B,C$ meet the opposite sides at $A_1,B_1,C_1$, respectively. Prove that if the quadrilateral $BA_1B_1C_1$ is cyclic, then
$$\frac{AC}{AB+BC}=\frac{AB}{AC+CB}+\frac{BC}{BA+AC}.$$