Let A, B be points lying on distinct sides of a line k. Find a circle o through A, B such that the length of the chord of o lying inside k is minimal.
Intuitively, it seems to me that the circle that will minimize the chord is the circle with points A and B lying on the same diameter (i.e. the segment AB is a diameter of the circle). I just do not know where to start or how to rigorously prove such a circle minimizes the chord.
Any help would be much appreciated.
Suppose the line $k$ and $AB$ meet at $E$. Let $C$ and $D$ be the points of intersection of $k$ with the variable circle through $A$ and $B$. We wish to minimize $CD$.
$CD=CE+ED\geq 2\sqrt{CE.ED}=2\sqrt{AE.EB}$. Equality is attained when $CE=ED$, that is when the center of the circle $O$ lies on the line perpendicular to $k$ through $E$( and $O$ must also lie on the perpendicular bisector of $AB$.)
$\blacksquare$