Found problems: 509
Circles $\omega_1$ and $\omega_2$ with centers at points $O_1$ and $O_2$ intersect at points $A$ and $B$. A point $C$ is constructed such that $AO_2CO_1$ is a parallelogram. An arbitrary line is drawn through point $A$, which intersects the circles $\omega_1$ and $\omega_2$ for the second time at points $X$ and $Y$, respectively. Prove that $CX = CY$.
(Oleksii Masalitin)
Given a semicirle with center $O$ an arbitrary inner point of the diameter divides it into two segments. Let there be semicircles above the two segments as visible in the below figure. The line $\ell$ passing through the point $A$ intersects the semicircles in $4$ points: $B, C, D$ and $E$. Show that the segments $BC$ and $DE$ have the same length.
[img]https://cdn.artofproblemsolving.com/attachments/1/4/86a369d54fef7e25a51fea6481c0b5e7dd45ff.png[/img]
$ABC$ is a triangle with $AB = 33$, $AC = 21$ and $BC = m$, an integer. There are points $D$, $E$ on the sides $AB$, $AC$ respectively such that $AD = DE = EC = n$, an integer. Find $m$.
Let $\ell$ be a straight line and $P \notin \ell$ be a point in the plane. On $\ell$ are, in this arrangement, points $A_1, A_2,...$ such that the radii of the incircles of all triangles $P A_iA_{i + 1}$ are equal. Let $k \in N$. Show that the radius of the incircle of the triangle $P A_j A_{j + k}$ does not depend on the choice of $j \in N$ .
Consider an equilateral triangle $\vartriangle ABC$ and points $D$ and $E$ on the sides $AB$ and $BC$csuch that $AE = CD$. Find the locus of intersection points of $AE$ with $CD$ as points $D$ and $E$ vary.
Let $I$ be the center of a circle inscribed in triangle $ABC$, in which $\angle BAC = 60 ^o$ and $AB \ne AC$. The points $D$ and $E$ were marked on the rays $BA$ and $CA$ so that $BD = CE = BC$. Prove that the line $DE$ passes through the point $I$.
In the acute-angled triangle $ABC$ all angles are greater than $45^o$. Let $AM$ and $BN$ be the heights of this triangle and let $X$ and $Y$ be the points on $MA$ and $NB$, respecively, such that $|MX| =|MB|$ and $|NY| =|NA|$. Prove that $MN$ and $XY$ are parallel.
On side $AB$ of triangle $ABC$, point $M$ is selected. A straight line passing through $M$ intersects the segment $AC$ at point $N$ and the ray $CB$ at point $K$. The circumscribed circle of the triangle $AMN$ intersects $\omega$, the circumscribed circle of the triangle $ABC$, at points $A$ and $S$. Straight lines $SM$ and $SK$ intersect with $\omega$ for the second time at points $P$ and $Q$, respectively. Prove that $AC = PQ$.
In triangle $ABC, \angle A = 90^o$. $M$ is the midpoint of $BC$ and $H$ is the foot of the altitude from $A$ to $BC$. The line passing through $M$ and perpendicular to $AC$ meets the circumcircle of triangle $AMC$ again at $P$. If $BP$ intersects $AH$ at $K$, prove that $AK = KH$.
Let $ABCDEF$ be a convex hexagon satisfying $AC = DF, CE = FB$ and $EA = BD$. Prove that the lines connecting the midpoints of opposite sides of the hexagon $ABCDEF$ intersect in one point.
A parallelogram $ABCD$ with $|AD| =|BD|$ has been given. A point $E$ lies on line segment $|BD|$ in such a way that $|AE| = |DE|$. The (extended) line $AE$ intersects line segment $BC$ in $F$. Line $DF$ is the angle bisector of angle $CDE$. Determine the size of angle $ABD$.
[asy]
unitsize (3 cm);
pair A, B, C, D, E, F;
D = (0,0);
A = dir(250);
B = dir(290);
C = B + D - A;
E = extension((A + D)/2, (A + D)/2 + rotate(90)*(A - D), B, D);
F = extension(A, E, B, C);
draw(A--B--C--D--cycle);
draw(A--F--D--B);
dot("$A$", A, SW);
dot("$B$", B, SE);
dot("$C$", C, NE);
dot("$D$", D, NW);
dot("$E$", E, S);
dot("$F$", F, SE);
[/asy]
$ABCD$ is a parallelogram. The excircle of $ABC$ opposite $A$ has center $E$ and touches the line $AB$ at $X$. The excircle of $ADC$ opposite $A$ has center $F$ and touches the line $AD$ at $Y$. The line $FC$ meets the line$ AB$ at $W$, and the line $EC$ meets the line $AD$ at $Z$. Show that $WX = YZ$.
Given two circles $\omega_1$ and $\omega_2$, with centers $O_1$ and $O_2$, respectively intesrecting at two points $A$ and $B$. Let $X$ and $Y$ be points on $\omega_1$. The lines $XA$ and $YA$ intersect $\omega_2$ again in $Z$ and $W$, respectively, such that $A$ is between $X,Z$ and $A$ is between $Y,W$. Let $M$ be the midpoint of $O_1O_2$, S be the midpoint of $XA$ and $T$ be the midpoint of $WA$. Prove that $MS = MT$ if, and only if, the points $X, Y, Z$ and $W$ are concyclic.
Given a chord $PQ$ of a circle and $M$ the midpoint of the chord, let $AB$ and $CD$ be two chords that pass through $M$. $AC$ and $BD$ are drawn until $PQ$ is intersected at points $X$ and $Y$ respectively. Show that $X$ and $Y$ are equidistant from $M$.
Let $Z$ and $Y$ be the tangency points of the incircle of the triangle $ABC$ with the sides $AB$ and $CA$, respectively. The parallel line to $Y Z$ through the midpoint $M$ of $BC$, meets $CA$ in $N$. Let $L$ be the point in $CA$ such that $NL = AB$ (and $L$ on the same side of $N$ than $A$). The line $ML$ meets $AB$ in $K$. Prove that $KA = NC$.
Let $ABCDE$ be a pentagon with right angles at vertices $B$ and $E$ and such that $AB = AE$ and $BC = CD = DE$. The diagonals $BD$ and $CE$ meet at point $F$. Prove that $FA = AB$.
Let $A,B,C,D$, be four different points on a line $\ell$, so that $AB=BC=CD$. In one of the semiplanes determined by the line $\ell$, the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points of the plane be such that the triangles $MAP$ and $NQD$ are equilateral (the vertices are also named clockwise). Find the angle $\angle MBN$.
From the vertex $A$ of a triangle $ABC$, three segments are drawn: the bisectors $AM$ and $AN$ of its interior and exterior angles and the tangent $AK$ to the circumscribed circle of the triangle (the points $M$, $K$ and $N$ lie on the line $BC$). Prove that $MK = KN$.
(I Sharygin)
Let $ABC$ be an acute triangle and let $M$, $N$, and $ P$ be on $CB$, $AC$, and $AB$, respectively, such that $AB = AN + PB$, $BC = BP + MC$, $CA = CM + AN$. Let $\ell$ be a line in a different half plane than $C$ with respect to to the line $AB$ such that if $X, Y$ are the projections of $A, B$ on $\ell$ respectively, $AX = AP$ and $BY = BP$. Prove that $NXYM$ is a cyclic quadrilateral.
In a circle with center $O$, chords $AB$ and $AC$ are drawn, both equal to the radius. Points $A_1$, $B_1$ and $C_1$ are projections of points $A, B$ and $C$, respectively, onto an arbitrary diameter $XY$. Prove that one of the segments $XB_1$, $OA_1$ and $C_1Y$ is equal to the sum of the other two.
(A. Shklover)
Altitudes $AA_1$ and $CC_1$ of an acute-angled triangle $ABC$ intersect at point $H$. A straight line passing through point $H$ parallel to line $A_1C_1$ intersects the circumscribed circles of triangles $AHC_1$ and $CHA_1$ at points $X$ and $Y$, respectively. Prove that points $X$ and $Y$ are equidistant from the midpoint of segment $BH$.
Let $ABC$ be a right triangle. It is known that there are points $D$ on the side $AC$ and $E$ on the side $BC$ such that $AB = AD = BE$ and $BD$ is perpendicular to $DE$. Calculate the ratios $\frac{AB}{BC}$ and $\frac{BC}{CA}$.
Let $ABCD$ be a quadrilateral with $AB = BC = CD = DA$.
(a) Prove that point A must be outside of triangle $BCD$.
(b) Prove that each pair of opposite sides on $ABCD$ is always parallel.
Let $ABC$ be a triangle inscribed in circle $(I)$ that is tangent to the sides $BC,CA,AB$ at points $D,E, F$ respectively. Assume that $L$ is the intersection of $BE$ and $CF,G$ is the centroid of triangle $DEF,K$ is the symmetric point of $L$ about $G$. If $DK$ meets $EF$ at $P, Q$ is on $EF$ such that $QF = PE$, prove that $\angle DGE + \angle FGQ = 180^o$.
Nguyễn Minh Hà
Let $ABC$ be a triangle such that $\angle CAB = 60^o$. Consider $D, E$ points on sides $AC$ and $AB$ respectively such that $BD$ bisects angle $\angle ABC$ , $CE$ bisects angle $\angle BCA$ and let $I$ be the intersection of them. Prove that $|ID| =|IE|$.