eutrigon and co-eutrigon

in some novel and interesting work on the relationships between parts to wholes, Wayne A Roberts (Canberra, see the 'principles of nature' work) has established Pythagoras-like relationships for triangles that have one 60 degree angle

he has named these 'eutrigons' and triangles with one 120 degree angle (which he refers to as 'co-eutrigons')

he utilises unit triangles on an isometric grid
see the section on isometric areas on this blog

the relationships are straightforwardly derived from the cosine rule, substituting 60 degrees and 120 degrees but, importantly as far as appreciation is concerned, he presents justifications based on diagrams:

a 'eutrigon' is a triangle with one 60 degree angle

the eutrigon has equilateral triangles drawn on each of the sides

what is the relationship between their areas?

the triangles that look as if they are equilateral are equilateral

a dynamic version, created by Wayne Roberts:

for the co-eutrigon, this diagram is offered by Wayne Roberts: