We will represent lines in terms of the two args of their limiting directions.

argPoint[arg_] := {Cos[arg], Sin[arg]}

PLine[arg1_, arg2_] := If[N[Cos[(arg2 - arg1) / 2]] == 0,
	Line[{argPoint[arg1], argPoint[arg2]}],
	If[N[arg2 - arg1 > Pi], PLine[arg2, arg1],
	If[N[arg2 - arg1] < 0, PLine[arg1, arg2 + 2 Pi],
	Circle[Sec[(arg2 - arg1) / 2] *
			argPoint[(arg1 + arg2) / 2],
			Tan[(arg2 - arg1) / 2],
			{arg2 + Pi/2, arg1 - Pi/2
				+ If[N[arg2 + Pi] > arg1, 2 Pi, 0]}]
	]]]

We make a simple function to draw PLines.

PShow[stuff_] := Show[Graphics[{Circle[{0,0},1],stuff}],
	AspectRatio -> Automatic]

PShow[Table[PLine[x, x+y], {x, 0, 2 Pi, .6},
	{y, .1, 6, .3}]]

Now we need a function to give the perpendicular to a given line through a given limit point (the answer can be given just as the other limit point of the new line).

PPerp[{arg1_, arg2_}, arg3_] :=
	2 Apply[ArcTan,
			intersection[line[argPoint[arg1], argPoint[arg2]],
						line[argPoint[arg3], arg3 + Pi/2]
						]] - arg3

Now we are doing Euclidean geometry.  We can have a line of two points, or a point and a direction.  We prefer the second form.

line[p1_List, p2_List] := line[p1, Apply[ArcTan, p2 - p1]]

intersection[line[{x1_, y1_}, a1_], line[{x2_, y2_}, a2_]] :=
	{x1, y1} + argPoint[a1] * (a /. Flatten[{
	NSolve[{x1 + a Cos[a1] == x2 + b Cos[a2],
			y1 + a Sin[a1] == y2 + b Sin[a2]}, {a, b}],
		a -> 10.^6}])

Off[Solve::luc]

Now we make a function to make tilings like Igor's.

PTree[a1_, {a2_, a3_}, lev_] := If[lev == 0,
	PLine[a2, a3],
	{PLine[a2, a3],
		PTree[a2, {PPerp[{a2, a3}, a1], a3}, lev - 1],
		PTree[a3, {a2, PPerp[{a2, a3}, a1]}, lev - 1]
	}]

trinest[{a1_, a2_, a3_}, lev_] := PShow[{
	PTree[a1, {a2, a3}, lev],
	PTree[a2, {a3, a1}, lev],
	PTree[a3, {a1, a2}, lev]}]

trinest[N[{0, 2 Pi / 3, 4 Pi / 3}], 5]

Now we make a nifty animation.

foo[x_] := 1 + (2 x - x ^ 2 + 2 x ^ 3) / 3

Do[trinest[N[{0, foo[x] Pi / 3, (x+1) 2 Pi / 3}], 5],
	{x, .025, .975, .05}]

Do[
Do[trinest[N[{0 - (x+rr) 2 Pi / 3,
				foo[x] Pi / 3 - (x+rr) 2 Pi / 3,
				(x+1) 2 Pi / 3 - (x+rr) 2 Pi / 3}], 5],
	{x, .025, .975, .05}],
	{rr, 0, 2}]