As the subject says, my questions are about the utility accessed from the menus: Extension > Render > Draw From Triangle... and specifically about whether one of the several 'circle' options is what I want, which is what I am thinking of, which is the following:

Nodes in a path are ordered, right? So an SVG renderer knows how to connect them up.
Therefore, the individual lines or Bézier curves (not B-splines, as a spline is multiple curves connected together at their ends, and having the same slopes [for Spiro, which is just a subset of B-splines, curvature also] at the points of those connections as their neighbors; polylines are not splines since they are meant not to have the same slope as their neighbors [or else two such connected lines could be replaced with a single line]) can also be ordered:
the curve between nodes 1 and 2 is curve 1, n2 & n3 = c2, and so on. If the path is closed, node last's connection to node 1 is the last curve.

So far so good (if all I have said above is technically correct).

I have happily drawn circumcircles and incircles around triangles before in Inkscape.
Geometrically speaking, they do not have to be triangles in the literal sense, since:

they can be closed or open
a quick test just now shows that using nonlinear sides for a three-sided, three-node closed shape produces a circumcircle (for instance, selected just for the test), although it neither encompasses the path nor contains the points where the nodes are at.

I tried, but three lone nodes (or any number of nodes not connected to any other) cannot be created in Inkscape, not sure about SVG itself.

Anyway, now I realised I wanted something else. Given three points, in either a closed or open triangle, I want a circle for which the two sides connected to the 'middle' node (they are ordered, remember) are both tangent to said circle: tangent at the first and last nodes, to be specific. There are infinitely many circles tangent to two true (that is, infinite) lines, see something like http://i.stack.imgur.com/2NFBD.png but without the intervening lines if you want proof :P but as far as I can tell, only one of those infinitely many circles fits the description I just typed, because either one of the sides connected to the middle node establishes a distance from said node, which selects a unique circle.

Is there a mathematical name for this circle in relation to the triangle that generates it that I simply have not learned yet? If so, is it already in 'Draw From Triangle'? If not, could it be? Another feature request in respect to that module could be informative error messages. Often it just fails silently, either producing no objects or something different from the description, especially when multiple desired shapes are requested.

In any case, if that ramble if readable, I would love some thoughts.

-Arlo James Barnes