Found problems: 45
Given an integer $n \ge 3$, prove that the diameter of a convex $n$-gon (interior and boundary) containing a disc of radius $r$ is (strictly) greater than $r(1 + 1/ \cos( \pi /n))$.
The Editors
A circle $\Gamma$ and a line $\ell$ which does not intersect $\Gamma$ are given. Suppose $P, Q,R, S$ are variable points on circle $\Gamma$ such that the points $A = PQ\cap RS$ and $B = PS \cap QR$ lie on $\ell$. Prove that the circle on $AB$ as a diameter passes through two fixed points.
Let $A$ and $B$ be two arbitrary points on the circumference of the circle such that $AB$ is not the diameter of the circle. The tangents to the circle drawn at points $A$ and $B$ meet at $T$. Next, choose the diameter $XY$ so that the segments $AX$ and $BY$ intersect. Let this be the intersection of $Q$. Prove that the points $A, B$, and $Q$ lie on a circle with center $T$.
Let $AB$ be the diameter of circle $\Gamma$ centred at $O$. Point $C$ lies on ray $\overrightarrow{AB}$. The line through $C$ cuts circle $\Gamma$ at $D$ and $E$, with point $D$ being closer to $C$ than $E$ is. $OF$ is the diameter of the circumcircle of triangle $BOD$. Next, construct $CF$, cutting the circumcircle of triangle $BOD$ at $G$. Prove that $O,A,E,G$ are concyclic.
(Possibly proposed by Pak Wono)
On a circle with diameter $AB$, lie points $C$ and $D$. $XY$ is the diameter passing through the midpoint $K$ of the chord $CD$. Point $M$ is the projection of point $X$ onto line $AC$, and point $N$ is the projection of point $Y$ on line $BD$. Prove that points $M, N$ and $K$ are collinear.
(A. Zaslavsky)
Segment $AB$ is the diameter of a circle. Points $C$ and $D$ lie on the circle. The rays $AC$ and $AD$ intersect the tangent to the circle at point $B$ at points $P$ and $Q$, respectively. Show that points $C, D, P$ and $Q$ lie on a circle.
$n$ diameters divide a disk into $2n$ equal sectors. $n$ of the sectors are coloured blue , and the other $n$ are coloured red (in arbitrary order) . Blue sectors are numbered from $1$ to $n$ in the anticlockwise direction, starting from an arbitrary blue sector, and red sectors are numbered from $1$ to $n$ in the clockwise direction, starting from an arbitrary red sector. Prove that there is a semi-disk containing sectors with all numbers from $1$ to $n$.
(V Proizvolov)
Given the acute triangle $ABC$. A diameter of the circumscribed circle of the triangle intersects the sides $AC$ and $BC$, dividing the side $BC$ in half. Show that the same diameter divides the side $AC$ in a ratio of $1: 3$, calculated from $A$, if and only if $\tan B = 2 \tan C$.
In the triangle $ABC, I$ is the center of the inscribed circle, point $M$ lies on the side of $BC$, with $\angle BIM = 90^o$. Prove that the distance from point $M$ to line $AB$ is equal to the diameter of the circle inscribed in triangle $ABC$
Given a circle and a fixed point $P$ not lying on it. Find the geometrical locus of the orthocenters of the triangles $ABP$, where $AB$ is the diameter of the circle.
Let \( H \) be the orthocenter of an acute triangle \( ABC \), and let \( AT \) be the diameter of the circumcircle of this triangle. Points \( X \) and \( Y \) are chosen on sides \( AC \) and \( AB \), respectively, such that \( TX = TY \) and \( \angle XTY + \angle XAY = 90^\circ \). Prove that \( \angle XHY = 90^\circ \).
[i]
Proposed by Matthew Kurskyi[/i]
Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.
The triangle $ABC$ is inscribed in the circle $S$ and $AB <AC$. The line containing $A$ and is perpendicular to $BC$ meets $S$ in $P$ ($P \ne A$). Point $X$ is on the segment $AC$ and the line $BX$ intersects $S$ in $Q$ ($Q \ne B$). Show that $BX = CX$ if, and only if, $PQ$ is a diameter of $S$.
Circles $w_1$ and $w_2$ with centers at points $O_1$ and $O_2$ respectively, intersect at points $A$ and $B$. A line passing through point $B$, intersects the circles $w_1$ and $w_2$ at points $C$ and $D$ other than $B$. Tangents to the circles $w_1$ and $w_2$ at points $C$ and $D$ intersect at point $E$. Line $EA$ intersects the circumscribed circle $w$ of triangle $AO_1O_2$ at point $F$. Prove that the length of the segment is $EF$ is equal to the diameter of the circle $w$.
(Vovchenko V., Plotnikov M.)
Circle $\Gamma$ has radius $10$, center $O$, and diameter $AB$. Point $C$ lies on $\Gamma$ such that $AC = 12$. Let $P$ be the circumcenter of $\vartriangle AOC$. Line $AP$ intersects $\Gamma$ at $Q$, where $Q$ is different from $A$. Then the value of $\frac{AP}{AQ}$ can be expressed in the form $\frac{m}{n}$, where m and n are relatively prime positive integers. Compute $m + n$.
Circles $\Omega_a$ and $\Omega_b$ are externally tangent at $D$, circles $\Omega_b$ and $\Omega_c$ are externally tangent at $E$, circles $\Omega_a$ and $\Omega_c$ are externally tangent at $F$. Let $P$ be an arbitrary point on $\Omega_a$ different from $D$ and $F$. Extend $PD$ to meet $\Omega_b$ again at $B$, extend $BE$ to meet $\Omega_c$ again at $C$ and extend $CF$ to meet $\Omega_a$ again at $A$. Show that $PA$ is a diameter of circle $\Omega_a$.
$AB$ and $CD$ are perpendicular diameters of a circle. $L$ is the tangent to the circle at $A$. $M$ is a variable point on the minor arc $AC$. The ray $BM, DM$ meet the line $L$ at $P$ and $Q$ respectively. Show that $AP\cdot AQ = AB\cdot PQ$.
Show how to construct the point $M$ which gives$ BQ$ parallel to $DP$.
If the lines $OP$ and $BQ$ meet at $N$ find the locus of $N$.
The lines $BP$ and $BQ$ meet the tangent at $D$ at $P'$ and $Q'$ respectively. Find the relation between $P'$ and $Q$'.
The lines $D$P and $DQ$ meet the line $BC$ at $P"$ and $Q"$ respectively. Find the relation between $P"$ and $Q"$.
The circles $C_1, C_2$ and $C_3$ are externally tangent in pairs (each tangent to other two externally). Let $M$ the common point of $C_1$ and $C_2, N$ the common point of $C_2$ and $C_3$ and $P$ the common point of $C_3$ and $C_1$. Let $A$ be an arbitrary point of $C_1$. Line $AM$ cuts $C_2$ in $B$, line $BN$ cuts $C_3$ in $C$ and line $CP$ cuts $C_1$ in $D$. Prove that $AD$ is diameter of $C_1$.
In the triangle $ABC , \angle A = \alpha, BC = a$. The inscribed circle touches the lines $AB$ and $AC$ at points $M$ and $P$. Find the length of the chord cut by the line $MP$ in a circle with diameter $BC$.
$25$ point in a plane and for all $3$ points, we find $2$ points such that this $2$ points' distance less than $1$ $cm$ . Prove that at least $13$ points in a circle of radius $1$ $cm$.
The diameter of a circle of radius $R$ is divided into $4$ equal parts. The point $M$ is taken on the circle. Prove that the sum of the squares of the distances from the point $M$ to the points of division (together with the ends of the diameter) does not depend on the choice of the point $M$. Calculate this sum.
A circle is inscribed in an trapezoid. Show that the diagonals of the trapezoid intersect at a point on the diameter of the circle perpendicular to the two parallel sides.
Let $AB$ be the diameter of a circle $k$ and let $E$ be a point in the interior of $k$. The line $AE$ intersects $k$ a second time in $C \ne A$ and the line $BE$ intersects $k$ a second time in $D \ne B$.
Show that the value of $AC \cdot AE+BD\cdot BE$ is independent of the choice of $E$.
$3k$ points are marked on the circumference. They divide it onto $3k$ arcs. Some $k$ of them have length $1$, other $k$ of them have length $2$, the rest $k$ of them have length $3$. Prove that some two of the marked points are the ends of one diameter.
Let $ABC$ be a triangle. Suppose that the perpendicular bisector of $BC$ meets the circle of diameter $AB$ at a point $D$ at the opposite side of $BC$ with respect to $A$, and meets the circle through $A, C, D$ again at $E$. Prove that $\angle ACE=\angle BCD$.
[i]Proposed by José Manuel Guerra and Victor Domínguez[/i]