There is a very large number of possible three way dissections that can be
performed. To limit this number I have restricted myself to looking for three
way dissections of the regular polygons and polygrams that can be solved in
twelve or fewer pieces.
Triangle — Square — Pentagon (10 pieces)
Triangle — Square — Hexagon (9 pieces)
Discovered by Harry Lindgren (1964).
Triangle — Square — Octagon (11 pieces)
Triangle — Square — Dodecagon (11 pieces)
Triangle — Square — Pentagram (12 pieces)
Triangle — Square — Hexagram (8 pieces)
The dissection is hingeable.
Triangle — Square — Octagram {8/2} (12 pieces)
Triangle — Square — Dodecagram {12/2} (11 pieces)
Triangle — Pentagon — Hexagon (12 pieces)
Triangle — Pentagon — Hexagram (11 pieces)
Triangle — Hexagon — Octagon (12 pieces)
Triangle — Hexagon — Dodecagon (11 pieces)
Triangle — Hexagon — Hexagram (11 pieces)
Triangle — Heptagon — Hexagram (12 pieces)
There is a small twelfth piece that can just be seen at the centre of the hexagram.
I’ve not yet found a way to remove this blemish.