Found problems: 821
Let $ABC$ be an acute triangle , with $AB \neq AC$ and denote its orthocenter by $H$ . The point $D$ is located on the side $BC$ and the circumcircles of the triangles $ABD$ and $ACD$ intersects for the second time the lines $AC$ , respectively $AB$ in the points $E$ respectively $F$. If we denote by $P$ the intersection point of $BE$ and $CF$ then show that $HP \parallel BC$ if and only if $AD$ passes through the circumcenter of the triangle $ABC$.
Circles $S,S_1,S_2$ are given in a plane. $S_1$ and $S_2$ touch each other externally, and both touch $S$ internally at $A_1$ and $A_2$ respectively. The common internal tangent to $S_1$ and $S_2$ meets $S$ at $P$ and $Q.$ Let $B_1$ and $B_2$ be the intersections of $PA_1$ and $PA_2$ with $S_1$ and $S_2$, respectively. Prove that $B_1B_2$ is a common tangent to $S_1,S_2$
Find the locus of the orthocentres (i.e. the point where three altitudes meet) of the triangles inscribed in a given circle .
(A. Andjans, Riga)
Let $S$ be a circle with centre $O$. A chord $AB$, not a diameter, divides $S$ into two regions $R_1$ and $R_2$ such that $O$ belongs to $R_2$. Let $S_1$ be a circle with centre in $R_1$, touching $AB$ at $X$ and $S$ internally. Let $S_2$ be a circle with centre in $R_2$, touching $AB$ at $Y$, the circle $S$ internally and passing through the centre of $S$. The point $X$ lies on the diameter passing through the centre of $S_2$ and $\angle YXO=30^o$. If the radius of $S_2$ is $100 $ then what is the radius of $S_1$?
There are six small circles in the figure with a radius of $1$ and tangent to a large circle and the sides of the $ABC$ of an equilateral triangle, where touch points are $K, L$ and $M$ respectively with the midpoints of sides $AB, BC$ and $AC$. Find the radius of the large circle and the side of the triangle $ABC$.
[img]https://cdn.artofproblemsolving.com/attachments/3/0/f858dcc5840759993ea2722fd9b9b15c18f491.png[/img]
Six small circles of radius $1$ are drawn so that they are all tangent to a larger circle, and two of them are tangent to sides $BC$ and $AD$ of a rectangle $ABCD$ at their midpoints $K$ and $L$. Each of the remaining four small circles is tangent to two sides of the rectangle. The large circle is tangent to sides $AB$ and $CD$ of the rectangle and cuts the other two sides. Find the radius of the large circle.
[img]https://cdn.artofproblemsolving.com/attachments/b/4/a134da78d709fe7162c48d6b5c40bd1016c355.png[/img]
$A$ and $B$ lie on a circle. $P$ lies on the minor arc $AB$. $Q$ and $R$ (distinct from $P$) also lie on the circle, so that $P$ and $Q$ are equidistant from $A$, and $P$ and $R$ are equidistant from $B$. Show that the intersection of $AR$ and $BQ$ is the reflection of $P$ in $AB$.
In circle with radius $10$, point $M$ is on chord $PQ$ such that $PM=5$ and $MQ=10$. Through point $M$ we draw chords $AB$ and $CD$, and points $X$ and $Y$ are intersection points of chords $AD$ and $BC$ with chord $PQ$ (see picture), respectively. If $XM=3$ find $MY$
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYy9kLzBiMmFmM2ViOGVmOTlmZDA5NGY2ZWY4MjM1YWI0ZDZjNjJlNzA1LnBuZw==&rn=Z2VvbWV0cmlqYS5wbmc=[/img]
Consider a square $ABCD$. Let $M$ be the midpoint of segment $AB$, $\Gamma_1$ be the circle tangent to $\overline{AD}$, $\overline{AM}$ and $\overline{MC}$ with radius $r > 0$ and let $\Gamma_2$ be the circle tangent to $\overline{AD}$, $\overline{DC}$ and $\overline{MC}$ with radius $R > 0$. Prove that $R =\frac{2r}{r+1}$.
Two non-coplanar circles in space are tangent at a point and have the same tangents at this point. Show that both circles lie on some sphere.
Two circles $ G_1$ and $ G_2$ intersect at two points $ M$ and $ N$. Let $ AB$ be the line tangent to these circles at $ A$ and $ B$, respectively, so that $ M$ lies closer to $ AB$ than $ N$. Let $ CD$ be the line parallel to $ AB$ and passing through the point $ M$, with $ C$ on $ G_1$ and $ D$ on $ G_2$. Lines $ AC$ and $ BD$ meet at $ E$; lines $ AN$ and $ CD$ meet at $ P$; lines $ BN$ and $ CD$ meet at $ Q$. Show that $ EP \equal{} EQ$.
Point $M$ is the midpoint of the side $AB$ of an acute triangle $ABC$. Circle with center $M$ passing through point $ C$, intersects lines $AC ,BC$ for the second time at points $P,Q$ respectively. Point $R$ lies on segment $AB$ such that the triangles $APR$ and $BQR$ have equal areas. Prove that lines $PQ$ and $CR$ are perpendicular.
A treasure is buried at some point on a round island with a radius of $1$ km. On the coast of the island there is a mathematician with a device that indicates the direction to the treasure when the distance to the treasure does not exceed $500$ m. In addition, the mathematician has a map of the island, on which he can record all his movements, perform measurements and geometric constructions. The mathematician claims that he has an algorithm for how to get to the treasure after walking less than $4$ km. Could this be true?
(B. Frenkin)
Let $\gamma_1, \gamma_2,\gamma_3 $ be three circles of unit radius which touch each other externally. The common tangent to each pair of circles are drawn (and extended so that they intersect) and let the triangle formed by the common tangents be $\triangle XYZ$ . Find the length of each side of $\triangle XYZ$
Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that the two points are always equidistant from $P.$
Let $ABC$ be a triangle with $AB = AC$. With center in a point of the side $BC$, the circle $S$ is constructed that is tangent to the sides $AB$ and $AC$. Let $P$ and $Q$ be any points on the sides $AB$ and $AC$ respectively, such that $PQ$ is tangent to $S$. Show that $PB \cdot CQ = \left(\frac{BC}{2}\right)^2$
The sides of a triangle are $25,39,$ and $40$. The diameter of the circumscribed circle is:
$ \textbf{(A)}\ \frac{133}{3}\qquad\textbf{(B)}\ \frac{125}{3}\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 41\qquad\textbf{(E)}\ 40 $
On the plane circles $k$ and $\ell$ are intersected at points $C$ and $D$, where circle $k$ passes through the center $L$ of circle $\ell$. The straight line passing through point $D$ intersects circles $k$ and $\ell$ for the second time at points $A$ and $B$ respectively in such a way that $D$ is the interior point of segment $AB$. Show that $AB = AC$.
$ABCD$ is a cyclic convex quadrilateral whose diagonals meet at $X$. The circle $(AXD)$ cuts $CD$ again at $V$ and the circle $(BXC)$ cuts $AB$ again at $U$, such that $D$ lies strictly between $C$ and $V$ and $B$ lies strictly between $A$ and $U$. Let $P\in AB\cap CD$.\\ \\
If $M$ is the intersection point of the tangents to $U$ and $V$ at $(UPV)$ and $T$ is the second intersection of circles $(UPV)$ and $(PAC)$, prove that $\angle PTM=90^o$.\\ \\
[i](Vlad Robu)[/i]
Let be given $n\ge 3$ distinct points in the plane. Is it always possible to find a circle which passes through three of the points and contains none of the remaining points
(a) inside the circle.
(b) inside the circle or on its boundary?
A semicircle with diameter $AB = 2r$ is divided into two sectors by an arbitrary radius. To each of the sectors a circle is inscribed. These two circles touch A$B$ at $S$ and $T$. Show that $ST \ge 2r(\sqrt{2}-1)$.
Two circles $ G_1$ and $ G_2$ intersect at two points $ M$ and $ N$. Let $ AB$ be the line tangent to these circles at $ A$ and $ B$, respectively, so that $ M$ lies closer to $ AB$ than $ N$. Let $ CD$ be the line parallel to $ AB$ and passing through the point $ M$, with $ C$ on $ G_1$ and $ D$ on $ G_2$. Lines $ AC$ and $ BD$ meet at $ E$; lines $ AN$ and $ CD$ meet at $ P$; lines $ BN$ and $ CD$ meet at $ Q$. Show that $ EP \equal{} EQ$.
Construct for the triangle $ABC$ a circle $S$ passing through the point $B$ and touching the line $CA$ at the point $A$, a circle $T$ passing through the point $C$ and touches the line $BA$ at the point $A$. The second intersection point of the circles $S$ and $T$ is denoted by $D$. The intersection point of the line $AD$ and the circumscribed circle $\Delta ABC$ is denoted by $E$. Prove that $D$ is the midpoint of the segment $AE$.
Let $AM$ and $AN$ be the tangents to a circle $\Gamma$ drawn from a point $A$ ($M$ and $N$ lie on the circle). A line passing through $A$ cuts $\Gamma$ at $B$ and $C$, with B between $A$ and $C$ such that $AB: BC = 2: 3$. If $P$ is the intersection point of $AB$ and $MN$, calculate the ratio $AP: CP$ .
Let $ABC$ be a triangle with $\angle ABC \neq 90^\circ$ and $AB$ its shortest side. Denote by $H$ the intersection of the altitudes of triangle $ABC$. Let $K$ be the circle through $A$ with centre $B$. Let $D$ be the other intersection of $K$ and $AC$. Let $K$ intersect the circumcircle of $BCD$ again at $E$. If $F$ is the intersection of $DE$ and $BH$, show that $BD$ is tangent to the circle through $D$, $F$, and $H$.