Prove that all circles are similar

State Converse of Intersecting Chords Theorem?

Each side of a square $A B C D$ has a length of $1$ unit. Points $P$ and $Q$ belong to $A B$ and $D A$ respectively. The perimeter of triangle $A P Q$ is $2$ units. What will be the angle of$P C Q$ ?

Prove that for two fixed points $A$ and $B$ and a real $r>0$, the locus of points $P$ such that $\frac{AP}{BP}=r$ is a circle.

Prove that if a point is marked on each side of a triangle $∆ABC$, then the three Miquel circles (each through a vertex and the two marked points on the adjacent sides) are concurrent at a point $M$ called the Miquel point.

Prove that if $A^{\text{'}},B^{\text{'}},C^{\text{'}}$ be the points on the sides $BC, CA, AB$ of $∆ABC$, respectively, then the circles $A{B}^{\text{'}}{C}^{\text{'}}, A^{\text{'}}B{C}^{\text{'}}$ and $A^{\text{'}}{B}^{\text{'}}C$ pass through a common point. (called the Miquel point)

Show that in a convex quadrilateral $ABCD$, if $AB+CD = BC + DA$, then we can inscribe a semicircle that is tangential to all $4$ sides.

A circle touches all the four sides of a quadrilateral $ABCD$. Prove that $AB+CD=BC+DA$

Which one of the following mixtures will act as buffer?