Here we see a triangle constructed with all three sides tangent to an external circle. Because the solid and dotted blue segments have a common point and are bounded by tangencies to a common circle, they are the same length; similarly, the solid and dotted red lines have the same length; therefore the perimeter of the triangle is equal to the sum of the segments between the apex (of cyan and magenta) and the external tangencies --- which two segments are also equal; in particular, the perimeter is determined by the apex angle and the circle radius.
By way of a converse, the envelope of triangles within fixed rays (as the cyan and the magenta, continued away from the apex) and having a fixed perimeter is a circle arc tangent to those fixed rays at a distance of half the fixed perimeter from their intersection.
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