If arcs $AXB$ and $CYD$ of a circle are congruent,find the ratio of $AB$ and $CD.$

Three girls Reshma,Salma and Mandip are playing a game by standing on a circle of radius $5 \, m$ drawn in a park. Reshma throws a ball to Salma,Salma to Mandip,Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is $6 \, m$ each,what is the distance between Reshma and Mandip?

In the same plane,two circles can have a maximum of ............ common chords.

If a pair of opposite sides of a cyclic quadrilateral are equal,prove that its diagonals are also equal.

Prove that two circles cannot intersect at more than two points.

Prove that among all the chords of a circle passing through a given point inside the circle,the one that is perpendicular to the diameter passing through the point is the smallest.