Find the point that partitions a directed segment between two points in a given ratio.
Determine whether a point lies on a segment before partitioning
Problem
Determine whether point \((3,2)\) lies on the segment from \((0,0)\) to \((9,6)\).
Big Picture
What this problem is really about
Segment membership requires more than lying on the supporting line. We’ll parameterize the finite segment from start to end, solve both coordinate equations for one common parameter, use agreement to establish collinearity, check that the parameter stays within the endpoint bounds, and substitute back to verify the test point.
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