TITLE: How to draw $4$ touching circles
QUESTION [6 upvotes]: I am trying to figure out a way for drawing $4$ touching circles like this:

Would appreciate help of any kind.

REPLY [2 votes]: You can actually do this starting with $A$, $B$, $C$, and $D$, and you still have a degree of freedom... but there's a restriction on the locations of these four points: they must form a cyclic quadrilateral.  (I'm not sure why, but it works out that way - perhaps because the perpendicular sectors all meet or something).
Once we have the points in place, it's easy: make segments $a = AB$, $b = BC$, $c = CD$, $d = DA$, and their perpendicular bisectors $a'$ etc.  place a point $E$ anywhere along $a'$ (and I do mean anywhere: if it's too close it will make other circles flip sides but tangency still holds!), draw a line $EB$ and get its intersection $F$ on $b'$, $FC$ intersects $c'$ at $G$, $GD$ intersects $d'$ at $H$.  Then $E$, $F$, $G$, $H$ are the centers of four circles that meet at $A$, $B$, $C$, and $D$.