Check if the cross product of b-a and c-a is0: that means all the points are collinear. If they are, check if c's coordinates are between a's and b's. Use either the x or the y coordinates, as long as a and b are separate on that axis (or they're the same on both).
def is_on(a, b, c):
"Return true iff point c intersects the line segment from a to b."
# (or the degenerate case that all 3 points are coincident)
return (collinear(a, b, c)
and (within(a.x, c.x, b.x) if a.x != b.x else
within(a.y, c.y, b.y)))
def collinear(a, b, c):
"Return true iff a, b, and c all lie on the same line."
return (b.x - a.x) * (c.y - a.y) == (c.x - a.x) * (b.y - a.y)
def within(p, q, r):
"Return true iff q is between p and r (inclusive)."
return p <= q <= r or r <= q <= p
This answer used to be a mess of three updates. The worthwhile info from them: Brian Hayes's chapter in Beautiful Code covers the design space for a collinearity-test function -- useful background. Vincent's answer helped to improve this one. And it was Hayes who suggested testing only one of the x or the y coordinates; originally the code had and in place of if a.x != b.x else.