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    <p>Check if the cross product of <code>b-a</code> and <code>c-a</code> is<code>0</code>: that means all the points are collinear. If they are, check if <code>c</code>'s coordinates are between <code>a</code>'s and <code>b</code>'s. Use either the x or the y coordinates, as long as <code>a</code> and <code>b</code> are separate on that axis (or they're the same on both).</p> <pre><code>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 &lt;= q &lt;= r or r &lt;= q &lt;= p </code></pre> <p>This answer used to be a mess of three updates. The worthwhile info from them: Brian Hayes's <a href="http://bit-player.org/bph-publications/BeautifulCode-2007-Hayes.pdf" rel="nofollow noreferrer">chapter</a> in <em>Beautiful Code</em> covers the design space for a collinearity-test function -- useful background. <a href="https://stackoverflow.com/questions/328107/how-can-you-determine-a-point-is-between-two-other-points-on-a-line-segment#328337">Vincent's answer</a> 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 <code>and</code> in place of <code>if a.x != b.x else</code>.</p>
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    1. POHow can you determine a point is between two other points on a line segment?
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    1. USDarius Bacon
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    1. COSince range checking faster would it be better to range check first then check for collinear if in bounding box.
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      1. USGrant M
 

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