Found problems: 3585
Let $M$ be the midpoint of the cathetus $AC$ of a right-angled triangle $ABC$ $(\angle C=90^{\circ})$. The perpendicular from $M$ to the bisector of angle $ABC$ meets $AB$ at point $N$. Prove that the circumcircle of triangle $ANM$ touches the bisector of angle $ABC$.
Proposed by:D.Shvetsov
Let $ ABCD$ be a convex quadrilateral with $ BA\neq BC$. Denote the incircles of triangles $ ABC$ and $ ADC$ by $ \omega_{1}$ and $ \omega_{2}$ respectively. Suppose that there exists a circle $ \omega$ tangent to ray $ BA$ beyond $ A$ and to the ray $ BC$ beyond $ C$, which is also tangent to the lines $ AD$ and $ CD$. Prove that the common external tangents to $ \omega_{1}$ and $\omega_{2}$ intersect on $ \omega$.
[i]Author: Vladimir Shmarov, Russia[/i]
Let $ABCD$ be a curcumscribed quadrilateral with incenter $I$, and let $O_{1}, O_{2}$ be the circumcenters of triangles $AID$ and $CID$. Prove that the circumcenter of triangle $O_{1}IO_{2}$ lies on the bisector of angle $ABC$
Let $ABC$ be a triangle. Let $BE$ and $CF$ be internal angle bisectors of $\angle B$ and $\angle C$
respectively with $E$ on $AC$ and $F$ on $AB$. Suppose $X$ is a point on the segment $CF$
such that $AX$ perpendicular $CF$; and $Y$ is a point on the segment $BE$ such that $AY$ perpendicular $BE$. Prove
that $XY = (b + c-a)/2$ where $BC = a, CA = b $and $AB = c$.
If in a quadrilateral $ABCD$ whose vertices lie on a circle of radius $1$, holds $$|AB| \cdot |BC| \cdot |CD|\cdot |DA| \ge 4$$, then $ABCD$ is a square. Prove it.
[hide=Hint given in contest] You can use Ptolemy's formula $|AB| \cdot |CD| + |BC|\cdot |AD|= |AC| \cdot|BD|$[/hide]
Let $ABC$ be a triangle with $m (\angle C) = 90^\circ$ and the points $D \in [AC], E\in [BC]$. Inside the triangle we construct the semicircles $C_1, C_2, C_3, C_4$ of diameters $[AC], [BC], [CD], [CE]$ and let $\{C, K\} = C_1 \cap C_2, \{C, M\} =C_3 \cap C_4, \{C, L\} = C_2 \cap C_3, \{C, N\} =C_1 \cap C_4$. Show that points $K, L, M, N$ are concyclic.
Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\Gamma.$ Let $E,F,G,H$ be the midpoints of arcs $AB,BC,CD,AD$ of $\Gamma,$ respectively. Suppose that $AC\cdot BD=EG\cdot FH.$ Show that $AC,BD,EG,FH$ are all concurrent.
Let $P_1P_2\dotsb P_{100}$ be a cyclic $100$-gon and let $P_i = P_{i+100}$ for all $i$. Define $Q_i$ as the intersection of diagonals $\overline{P_{i-2}P_{i+1}}$ and $\overline{P_{i-1}P_{i+2}}$ for all integers $i$.
Suppose there exists a point $P$ satisfying $\overline{PP_i}\perp\overline{P_{i-1}P_{i+1}}$ for all integers $i$. Prove that the points $Q_1,Q_2,\dots, Q_{100}$ are concyclic.
[i]Michael Ren[/i]
Let $ABC$ be a triangle and $\omega$ its incircle. $\omega$ touches $AC,AB$ at $E,F$, respectively. Let $P$ be a point on $EF$. Let $\omega_1=(BFP), \omega_2=(CEP)$. The parallel line through $P$ to $BC$ intersects $\omega_1,\omega_2$ at $X,Y$, respectively. Show that $BX=CY$.
For a triangle $T$ and a line $d$, if the feet of perpendicular lines from a point in the plane to the edges of $T$ all lie on $d$, say $d$ focuses $T$. If the set of lines focusing $T_1$ and the set of lines focusing $T_2$ are the same, say $T_1$ and $T_2$ are equivalent. Prove that, for any triangle in the plane, there exists exactly one equilateral triangle which is equivalent to it.
Let $D$ and $E$ be points in the interiors of sides $AB$ and $AC$, respectively, of a triangle $ABC$, such that $DB = BC = CE$. Let the lines $CD$ and $BE$ meet at $F$. Prove that the incentre $I$ of triangle $ABC$, the orthocentre $H$ of triangle $DEF$ and the midpoint $M$ of the arc $BAC$ of the circumcircle of triangle $ABC$ are collinear.
In triangle $ABC$ with $CA=CB$, point $E$ lies on the circumcircle of $ABC$ such that $\angle ECB=90^{\circ}$. The line through $E$ parallel to $CB$ intersects $CA$ in $F$ and $AB$ in $G$. Prove that the center of the circumcircle of triangle $EGB$ lies on the circumcircle of triangle $ECF$.
Proposed by Prithwijit De
Let $ABC$ be a triangle in which no angle is $90^{\circ}$. For any point $P$ in the plane of the triangle, let $A_1, B_1, C_1$ denote the reflections of $P$ in the sides $BC,CA,AB$ respectively. Prove that
(i) If $P$ is the incenter or an excentre of $ABC$, then $P$ is the circumenter of $A_1B_1C_1$;
(ii) If $P$ is the circumcentre of $ABC$, then $P$ is the orthocentre of $A_1B_1C_1$;
(iii) If $P$ is the orthocentre of $ABC$, then $P$ is either the incentre or an excentre of $A_1B_1C_1$.
Peter is a pentacrat and spends his time drawing pentagrams.
With the abbreviation $\left|XYZ\right|$ for the area of an arbitrary triangle $XYZ$, he notes that any convex pentagon $ABCDE$ satisfies the equality
$\left|EAC\right|\cdot\left|EBD\right|=\left|EAB\right|\cdot\left|ECD\right|+\left|EBC\right|\cdot\left|EDA\right|$.
Guess what you are supposed to do and do it.
A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.
Let $ABC$ be a triangle with $BC=13, CA=11, AB=10$. Let $A_1$ be the midpoint of $BC$. A variable line $\ell$ passes through $A_1$ and meets $AC,AB$ at $B_1,C_1$. Let $B_2,C_2$ be points with $B_2B=B_2C, B_2C_1\perp AB, C_2B=C_2C, C_2B_1 \perp AC$, and define $P=BB_2\cap CC_2$. Suppose the circles of diameters $BB_2, CC_2$ meet at a point $Q\neq A_1$. Given that $Q$ lies on the same side of line $BC$ as $A$, the minimum possible value of $\dfrac{PB}{PC}+\dfrac{QB}{QC}$ can be expressed in the form $\dfrac{a\sqrt{b}}{c}$ for positive integers $a,b,c$ with $\gcd (a,c)=1$ and $b$ squarefree. Determine $a+b+c$.
[i]Proposed by Vincent Huang[/i]
Suppose $ABCD$ is a cyclic quadrilateral inscribed in a circle of radius one unit. If $AB \cdot BC \cdot CD \cdot DA \geq 4$, prove that $ABCD$ is a square.
In $\triangle ABC$, points $D$, $E$, and $F$ lie on sides $BC$, $CA$, and $AB$, respectively, such that each of the quadrilaterals $AFDE$, $BDEF$, and $CEFD$ has an incircle. Prove that the inradius of $\triangle ABC$ is twice the inradius of $\triangle DEF$.
Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$. The lines $AB$ and $CD$ meet at $P$, the lines $AD$ and $BC$ meet at $Q$, and the diagonals $AC$ and $BD$ meet at $R$. Let $M$ be the midpoint of the segment $PQ$, and let $K$ be the common point of the segment $MR$ and the circle $\omega$. Prove that the circumcircle of the triangle $KPQ$ and $\omega$ are tangent to one another.
Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic.
[i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]
Let $ABCD$ be a cyclic quadrilateral; $M_{ac}$ be the midpoint of $AC$; $H_d,H_b$ be the orthocenters of $\triangle ABC,\triangle ADC$ respectively; $P_d,P_b$ be the projections of $H_d$ and $H_b$ to $BM_{ac}$ and $DM_{ac}$ respectively. Define similarly $P_a,P_c$ for the diagonal $BD$. Prove that $P_a,P_b,P_c,P_d$ are concyclic.
Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB=BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \neq B$. Prove that the lines $BB_1$ and $DE$ intersect on $\omega$.
Let $O$ denote the circumcentre of an acute-angled triangle $ABC$. Let point $P$ on side $AB$ be such that $\angle BOP = \angle ABC$, and let point $Q$ on side $AC$ be such that $\angle COQ = \angle ACB$. Prove that the reflection of $BC$ in the line $PQ$ is tangent to the circumcircle of triangle $APQ$.
Let $\Delta ABC$ be a triangle with circumcenter $O$ and circumcircle $w$. Let $S$ be the center of the circle which is tangent with $AB$, $AC$, and $w$ (in the inside), and let the circle meet $w$ at point $K$. Let the circle with diameter $AS$ meet $w$ at $T$. If $M$ is the midpoint of $BC$, show that $K,T,M,O$ are concyclic.
Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$ whose diagonals meet at $F$. Lines $AB$ and $CD$ meet at $E$. Segment $EF$ intersects $\omega$ at $X$. Lines $BX$ and $CD$ meet at $M$, and lines $CX$ and $AB$ meet at $N$. Prove that $MN$ and $BC$ concur with the tangent to $\omega$ at $X$.
[i]Proposed by Allen Liu[/i]