I'll be the first to say I'm not good at geometry. May be a simpler way.

This does seem to match your blue curve. Coincidence? Will it work in all cases? Don't know :)

Draw two small half circles (magenta)

Draw straight line to ends of circles (green)

Select both circles and Blend [Tangent G1, Bulge = 1] (cyan)

Pilou find apex trick (orange lines)

Draw Perp to Perp line from Blend apex to green line (black)

Draw 3-point arc from top to bottom of green line while snapping the center to the intersection of the black & red lines (blue)

Ed Ferguson