Equations in trilinear coordinates for the line through two points, and the circle through three 
I am looking for the equations which describe, using trilinear coordinates:

*

*A line through two given points.

*A circle through three given non-collinear points.


I would also love to get a refernce for a good source on the subject.
Thank you!
 A: Trilinear coordinates are a special case of homogeneous coordiates. So if a point $p$ has trilinear coordinates $[x:y:z]$ then you describe a line of such points by a homogeneous equation $ax+by+cz=0$ or equivalently by a coordinate vector $g=[a:b:c]$. Incidence is checked by the dot product, so $p$ lies on $g$ iff $\langle g,p\rangle=ax+by+cz=0$. Given two points $p,q$ the line spanned by them has coordinate vector $p\times q$. See What is the interpretation of homogeneous line intersection? for further discussion.
Circles are harder. Homogeneous coordinates usually go hand in hand with projective geometry, and a generic conic is a much more natural concept there than a circle. You can't find a description of the circle without taking the shape of the basis triangle into account in some way.
In the usual homogenization of the Euclidean plane using $(x,y)\mapsto[x:y:1]$ all circles pass through the ideal circle points $[1:\pm i:0]$. A conic is defined by five points, so three points plus these circle points define a circle. Switching to trilinear coordinates is a change of basis, so the circle points still exist as a concept, but their coordinates are different and more complicated.
A change of basis is equivalent to a projective transformation, which is uniquely defined by four points and their images, as detailed here. So by mapping e.g. the three corners and the incircle center $[1:1:1]$ you could find a matrix for this change of coordinates. You could apply it to the ideal circle points, then describe the circle as a conic through these.
