Found problems: 114
Point $A$ is located in this circle of radius $1$. An arbitrary chord is drawn through it, and then a circle of radius $2$ is drawn through the ends of this chord. Prove that all such circles touch some fixed circle, not depending from the initial choice of the chord.
Let $\Gamma_1$ be a circle with centre $A$ and $\Gamma_2$ be a circle with centre $B$, with $A$ lying on $\Gamma_2$. On $\Gamma_2$ there is a (variable) point $P$ not lying on $AB$. A line through $P$ is a tangent of $\Gamma_1$ at $S$, and it intersects $\Gamma_2$ again in $Q$, with $P$ and $Q$ lying on the same side of $AB$. A different line through $Q$ is tangent to $\Gamma_1$ at $T$. Moreover, let $M$ be the foot of the perpendicular to $AB$ through $P$. Let $N$ be the intersection of $AQ$ and $MT$.
Show that $N$ lies on a line independent of the position of $P$ on $\Gamma_2$.
In a triangle $ABC, AC = BC$ and $D$ is the midpoint of $AB$. Let $E$ be an arbitrary point on line $AB$ which is not $B$ or $D$. Let $O$ be the circumcenter of $\vartriangle ACE$ and $F$ the intersection of the perpendicular from $E$ to $BC$ and the perpendicular to $DO$ at $D$. Prove that the acute angle between $BC$ and $BF$ does not depend on the choice of point $E$.
An arbitrary point $P$ is chosen on the lateral side $AB$ of the trapezoid $ABCD$. Straight lines passing through it parallel to the diagonals of the trapezoid intersect the bases at points $Q$ and $R$. Prove that the sides $QR$ of all possible triangles $PQR$ pass through a fixed point.
Let be given a circle with center $O$ and a point $P$ outside the circle. A line $l$ passes through $P$ and cuts the circle at $A$ and $B$. Let $C$ be the point symmetric to $A$ with respect to $OP$, and let $m$ be the line $BC$. Prove that all lines $m$ have a common point as $l$ varies.
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.
Isosceles triangles with a fixed angle $\alpha$ at the vertex opposite to the base are being inscribed into a rectangle $ABCD$ so that this vertex lies on the side $BC$ and the vertices of the base lie on the sides $AB$ and $CD$. Prove that the midpoints of the bases of all such triangles coincide.
(Igor Zhizhilkin)
Point $A$ is located in this circle of radius $1$. An arbitrary chord is drawn through it, and then a circle of radius $2$ is drawn through the ends of this chord. Prove that all such circles touch some fixed circle, not depending from the initial choice of the chord.
Let $h$ be a semicircle with diameter $AB$. Let $P$ be an arbitrary point inside the diameter $AB$. The perpendicular through $P$ on $AB$ intersects $h$ at point $C$. The line $PC$ divides the semicircular area into two parts. A circle will be inscribed in each of them that touches $AB, PC$ and $h$. The points of contact of the two circles with $AB$ are denoted by $D$ and $E$, where $D$ lies between $A$ and $P$. Prove that the size of the angle $DCE$ does not depend on the choice of $P$.
(Walther Janous)
Prove that if the point $ P $ runs through a circle inscribed in the triangle $ ABC $, then the value of the expression
$ a \cdot PA^2 + b \cdot PB^2 + c \cdot PC^2 $ is constant ($ a, b, c $ are the lengths of the sides opposite the vertices $ A, B, C $, respectively).
Let $ABC$ be a triangle and let $P$ be a point in the interior of the side $BC$. Let $I_1$ and $I_2$ be the incenters of the triangles $AP B$ and $AP C$, respectively. Let $X$ be the closest point to $A$ on the line $AP$ such that $XI_1$ is perpendicular to $XI_2$. Prove that the distance $AX$ is independent of the choice of $P$.
Two rectangles on a plane intersect at eight points. Consider every other intersection point, they are connected with line segments, these segments form a quadrilateral. Prove that the area of this quadrilateral does not vary under translations of one of the rectangles.
Let $ABC$ be an equilateral triangle such that length of its altitude is $1$. Circle with center on the same side of line $AB$ as point $C$ and radius $1$ touches side $AB$. Circle rolls on the side $AB$. While the circle is rolling, it constantly intersects sides $AC$ and $BC$. Prove that length of an arc of the circle, which lies inside the triangle, is constant
On the side $BC$ of the equilateral triangle $ABC$, choose any point $D$, and on the line $AD$, take the point $E$ such that $| B A | = | BE |$. Prove that the size of the angle $AEC$ is of does not depend on the choice of point $D$, and find its size.
Let $ABC$ be a triangle and let $P$ be a point in the interior of the side $BC$. Let $I_1$ and $I_2$ be the incenters of the triangles $AP B$ and $AP C$, respectively. Let $X$ be the closest point to $A$ on the line $AP$ such that $XI_1$ is perpendicular to $XI_2$. Prove that the distance $AX$ is independent of the choice of $P$.
The circle divides each side of an equilateral triangle into three equal parts. Prove that the sum of the squares of the distances from any point of this circle to the vertices of the triangle is constant.
Let $ABC$ be an isosceles triangle with apex $A$ and altitude $AD$. On $AB$, choose a point $F$ distinct from $B$ such that $CF$ is tangent to the incircle of $ABD$. Suppose that $\vartriangle BCF$ is isosceles. Show that those conditions uniquely determine:
a) which vertex of $BCF$ is its apex,
b) the size of $\angle BAC$
As shown in figure , $A,B$ are two fixed points of circle $\odot O$, $C$ is the midpoint of the major arc $AB$, $D$ is any point of the minor arc $AB$. Tangent at $D$ intersects tangents at $A,B$ at points $E,F$ respectively. Segments $CE$ and $CF$ intersect chord $AB$ at points $G$ and $H$ respectively. Prove that the length of line segment $GH$ has a fixed value.
[img]https://cdn.artofproblemsolving.com/attachments/9/2/85227f169193f61e313293e9128f6ece2ff1f7.png[/img]
Let $ABC$ be an acute triangle. Let $P$ be a point such that $PB$ and $PC$ are tangent to circumcircle of $ABC$. Let $X$ and $Y$ be variable points on $AB$ and $AC$, respectively, such that $\angle XPY = 2\angle BAC$ and $P$ lies in the interior of triangle $AXY$. Let $Z$ be the reflection of $A$ across $XY$. Prove that the circumcircle of $XYZ$ passes through a fixed point.
(Dominik Burek, Poland)
Let $ABC$ be an isosceles triangle ($AB=AC$). An arbitrary point $M$ is chosen on the extension of the $BC$ beyond point $B$. Prove that the sum of the radius of the circle inscribed in the triangle $AMB$ and the radius of the circle tangent to the side $AC$ and the extensions of the sides $AM, CM$ of the triangle $AMC$ does not depend on the choice of point $M$.
Given is a triangle $ABC$ with its circumscribed circle and $| AC | <| AB |$. On the short arc $AC$, there is a variable point $D\ne A$. Let $E$ be the reflection of $A$ wrt the inner bisector of $\angle BDC$. Prove that the line $DE$ passes through a fixed point, regardless of point $D$.
Let $ABC$ be an acute triangle and $D \in (BC) , E \in (AD)$ be mobile points. The circumcircle of triangle $CDE$ meets the median from $C$ of the triangle $ABC$ at $F$ Prove that the circumcenter of triangle $AEF$ lies on a fixed line.
An acute triangle $ABC$ is given. Let us denote $X$ for each of its inner points $X_a, X_b, X_c$ its images in axial symmetries sequentially along the lines $BC, CA, AB$. Prove that all $X_aX_bX_c$ triangles have a common interior point.
(Josef Tkadlec)
Let $ABC$ be a fixed triangle. Point $D$ is an arbitrary point on the side $BC$. Point $P$ is fixed on $AD$. The circumcircle of triangle $BPD$ meets $AB$ at $E$ distinct from $B$. Point $Q$ varies on $AP$. Let $BQ$ and $CQ$ meet the circumcircles of triangles $BPD, CPD$ respectively at $F,Z$ distinct from $B,C$. Prove that the circumcircle $EFZ$ is through a fixed point distinct from $E$ and this fixed point is on the circumcircle of triangle $CPD$.
Kostas Vittas
Point $A$ is located in this circle of radius $1$. An arbitrary chord is drawn through it, and then a circle of radius $2$ is drawn through the ends of this chord. Prove that all such circles touch some fixed circle, not depending from the initial choice of the chord.