Found problems: 3585
In the triangle $ABC$, variable points $D, E, F$ are on the sides[lines] $BC, CA, AB$ respectively so the triangle $DFE$ is similar to the triangle $ABC$ in this order. Circumcircles of $BDF$ and $CDE$ intersect respectively the circumcircle of $ABC$ at $P$ and $Q$ for the second time. Prove that the circumcircle of $DPQ$ passes through a fixed point.
Circles $C(O_1)$ and $C(O_2)$ intersect at points $A$, $B$. $CD$ passing through point $O_1$ intersects $C(O_1)$ at point $D$ and tangents $C(O_2)$ at point $C$. $AC$ tangents $C(O_1)$ at $A$. Draw $AE \bot CD$, and $AE$ intersects $C(O_1)$ at $E$. Draw $AF \bot DE$, and $AF$ intersects $DE$ at $F$. Prove that $BD$ bisects $AF$.
Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.
Let $I$ be the incenter of triangle $ABC$, and let $\omega$ be its incircle. Let $E$ and $F$ be the points of tangency of $\omega$ with $CA$ and $AB$, respectively. Let $X$ and $Y$ be the intersections of the circumcircle of $BIC$ and $\omega$. Take a point $T$ on $BC$ such that $\angle AIT$ is a right angle. Let $G$ be the intersection of $EF$ and $BC$, and let $Z$ be the intersection of $XY$ and $AT$. Prove that $AZ$, $ZG$, and $AI$ form an isosceles triangle.
[i]Proposed by Li4 and usjl.[/i]
Let $ABCD$ be any convex quadrilateral and let $P, Q, R, S$ be points on the segments $AB, BC, CD$, and $DA$, respectively. It is given that the segments $PR$ and $QS$ dissect $ABCD$ into four quadrilaterals, each of which has perpendicular diagonals. Show that the points $P, Q, R, S$ are concyclic.
A quadrilateral \( ABCD \) with no parallel sides is inscribed in a circle \( \Omega \). Circles \( \omega_a, \omega_b, \omega_c, \omega_d \) are inscribed in triangles \( DAB, ABC, BCD, CDA \), respectively. Common external tangents are drawn between \( \omega_a \) and \( \omega_b \), \( \omega_b \) and \( \omega_c \), \( \omega_c \) and \( \omega_d \), and \( \omega_d \) and \( \omega_a \), not containing any sides of quadrilateral \( ABCD \). A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle \( \Gamma \). Prove that the lines joining the centers of \( \omega_a \) and \( \omega_c \), \( \omega_b \) and \( \omega_d \), and the centers of \( \Omega \) and \( \Gamma \) all intersect at one point. \\
Let $a > b > c > d$ be positive integers and suppose that \[ ac + bd = (b+d+a-c)(b+d-a+c). \] Prove that $ab + cd$ is not prime.
A hexagon $A_1A_2A_3A_4A_5A_6$ has no four concyclic vertices, and its diagonals $A_1A_4$, $A_2A_5$ and $A_3A_6$ concur. Let $l_i $ be the radical axis of circles $A_iA_{i+1}A_{i-2} $ and $A_iA_{i-1}A_{i+2} $ (the points $A_i $ and $A_{i+6} $ coincide). Prove that $l_i, i=1,\cdots,6$, concur.
Two lines $m$ and $n$ intersect in a unique point $P$. A point $M$ moves along $m$ with constant speed, while another point $N$ moves along $n$ with the same speed. They both pass through the point $P$, but not at the same time.
Show that there is a fixed point $Q \ne P$ such that the points $P,Q,M$ and $N$ lie on a common circle all the time.
Let $ABC$ be an acute triangle with orthocenter $H$. A point $L \ne A$ lies on the plane of $ABC$ such that $\overline{HL} \perp \overline{AL}$ and $LB : LC = AB : AC$. Suppose $M_1 \ne B$ lies on $\overline{BL}$ such that $\overline{HM_1} \perp \overline{BM_1}$ and $M_2 \ne C$ lies on $\overline{CL}$ such that $\overline{HM_2} \perp \overline{CM_2}$. Prove that $\overline{M_1M_2}$ bisects $\overline{AL}$.
Let $ABCD$ be a cyclic quadrilateral with $\angle ABC = 60^o$ and $| BC | = | CD |$. Prove that $|CD| + |DA| = |AB|$
Let $ABC$ be a triangle with circumcircle $\omega$ and let $\Omega_A$ be the $A$-excircle. Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_A$. Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_A$ at $X$ and $Y$ respectively. The tangent line at $P$ to the circumcircle of the triangle $APX$ intersects the tangent line at $Q$ to the circumcircle of the triangle $AQY$ at a point $R$. Prove that $\overline{AR} \perp \overline{BC}$.
Let $ABC$ ($BC > AC$) be an acute triangle with circumcircle $k$ centered at $O$. The tangent to $k$ at $C$ intersects the line $AB$ at the point $D$. The circumcircles of triangles $BCD, OCD$ and $AOB$ intersect the ray $CA$ (beyond $A$) at the points $Q, P$ and $K$, respectively, such that $P \in (AK)$ and $K \in (PQ)$. The line $PD$ intersects the circumcircle of triangle $BKQ$ at the point $T$, so that $P$ and $T$ are in different halfplanes with respect to $BQ$. Prove that $TB = TQ$.
Let $ABCD$ be a cyclic quadrilateral. A circle passing through $A$ and $B$ meets $AC$ and $BD$ at points $E$ and $F$ respectively. The lines $AF$ and $BC$ meet at point $P$, and the lines $BE$ and $AD$ meet at point $Q$. Prove that $PQ$ is parallel to $CD$.
Cyclic quadrilateral $ABCD$ with circumcenter $O$ is given. Point $P$ is the intersection of diagonals $AC$ and $BD$. Let $M$ and $N$ be the midpoint of the sides $AD$ and $BC$, respectively. Suppose that $\omega_1$, $\omega_2$ and $\omega_3$ be the circumcircle of triangles $ADP$, $BCP$ and $OMN$, respectively. The intersection point of $\omega_1$ and $\omega_3$, which is not on the arc $APD$ of $\omega_1$, is $E$ and the intersection point of $\omega_2$ and $\omega_3$, which is not on the arc $BPC$ of $\omega_2$, is $F$. Prove that $OF=OE$.
Proposed by Seyed Amirparsa Hosseini Nayeri
In the acute-angled triangle $ABC$, let $CD, AE$ be the altitudes. Points $F$ and $G$ are the projections of $A$ and $C$ on the line $DE$, respectively, $H$ and $K$ are the projections of $D$ and $E$ on the line $AC$, respectively. The lines $HF$ and $KG$ intersect at point $P$. Prove that line $BP$ bisects the segment $DE$.
In triangle $ABC$ we havev $\angle A=\frac{\pi}{3}$. Construct $E$ and $F$ on continue of $AB$ and $AC$ respectively such that $BE=CF=BC$. Suppose that $EF$ meets circumcircle of $\triangle ACE$ in $K$. ($K\not \equiv E$). Prove that $K$ is on the bisector of $\angle A$.
Quadrilateral $ABCD$ is inscribed in a circle. Let $\omega_A$, $\omega_B$, $\omega_C$, $\omega_D$ be the incircles of triangles $DAB$, $ABC$, $BCD$, $CDA$ respectively. The common external common tangent of $\omega_A$, $\omega_B$, different from line $AB$, meets the external common tangent of $\omega_A$, $\omega_D$, different from $AD$, at point $A'$. Similarly, the external common tangent of $\omega_B$, $\omega_C$ different from $BC$ meets the external common tangent of $\omega_C$, $\omega_D$ different from $CD$ at $C'$.
Prove that $AA'\parallel CC'$.
Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.
Let $ABCD$ be a square with side length $1$.
Find the locus of all points $P$ with the property $AP\cdot CP + BP\cdot DP = 1$.
A point $P$ is chosen inside a triangle $ABC$ with circumcircle $\Omega$. Let $\Gamma$ be the circle passing
through the circumcenters of the triangles $APB$, $BPC$, and $CPA$. Let $\Omega$ and $\Gamma$ intersect at
points $X$ and $Y$. Let $Q$ be the reflection of $P$ in the line $XY$ . Prove that $\angle BAP = \angle CAQ$.
Let $C_1, A_1, B_1$ be points on sides $AB, BC, CA$ of triangle $ABC$, such that $AA_1, BB_1, CC_1$ concur. The rays $B_1A_1$ and $B_1C_1$ meet the circumcircle of the triangle at points $A_2$ and $C_2$ respectively. Prove that $A, C$, the common point of $A_2C_2$ and $BB_1$ and the midpoint of $A_2C_2$ are concyclic.
Let $ABC$ be an acute scalene triangle, with the feet of $A,B,C$ onto $BC,CA,AB$ being $D,E,F$ respectively. Let $W$ be a point inside $ABC$ whose reflections over $BC,CA,AB$ are $W_a,W_b,W_c$ respectively. Finally, let $N$ and $I$ be the circumcenter and the incenter of $W_aW_bW_c$ respectively. Prove that, if $N$ coincides with the nine-point center of $DEF$, the line $WI$ is parallel to the Euler line of $ABC$.
[i]Proposed by Navneel Singhal, India and Massimiliano Foschi, Italy[/i]
Two circles X and Y in the plane intersect at two distinct points A and B such that the centre of Y lies on X. Let points C and D be on X and Y respectively, so that C, B and D are collinear. Let point E on Y be such that DE is parallel to AC. Show that AE = AB.
Let $N$ be the midpoint of arc $ABC$ of the circumcircle of triangle $ABC$, let $M$ be the midpoint of $AC$ and let $I_1, I_2$ be the incentres of triangles $ABM$ and $CBM$. Prove that points $I_1, I_2, B, N$ lie on a circle.
[i]M. Kungojin[/i]