Školjka

%V0 Given $n>3$ points in the plane such that no three of the points are collinear. Does there exist a circle passing through (at least) $3$ of the given points and not containing any other of the $n$ points in its interior ?

%V0 Given four points $A,$ $B,$ $C,$ $D$ on a circle such that $AB$ is a diameter and $CD$ is not a diameter. Show that the line joining the point of intersection of the tangents to the circle at the points $C$ and $D$ with the point of intersection of the lines $AC$ and $BD$ is perpendicular to the line $AB.$

%V0 Given a point $P$ lying on a line $g,$ and given a circle $K.$ Construct a circle passing through the point $P$ and touching the circle $K$ and the line $g.$

%V0 Show that the four perpendiculars dropped from the midpoints of the sides of a cyclic quadrilateral to the respective opposite sides are concurrent. Note by Darij: A cyclic quadrilateral is a quadrilateral inscribed in a circle.

%V0 Consider a circle with center $O$ and radius $R,$ and let $A$ and $B$ be two points in the plane of this circle. a.) Draw a chord $CD$ of the circle such that $CD$ is parallel to $AB,$ and the point of the intersection $P$ of the lines $AC$ and $BD$ lies on the circle. b.) Show that generally, one gets two possible points $P$ ($P_{1}$ and $P_{2}$) satisfying the condition of the above problem, and compute the distance between these two points, if the lengths $OA=a,$ $OB=b$ and $AB=d$ are given.

%V0 Describe all closed bounded figures $\Phi$ in the plane any two points of which are connectable by a semicircle lying in $\Phi$.