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# fourth point: two thirds in each direction from the first point
d = (x1 + 2. * (x2 - x1)/3., y1 + 2. * (y2 - y1)/3.)

# the last point
e = end

return {'a': a, 'b': b, 'c': c, 'd' : d, 'e' : e, 'factor' : factor}