Why the locus is a line

Remark — Why the squared coordinates always simplify

The equality of two distances always gives an equation of the type $(x-a_1)^2 + (y-b_1)^2 = (x-a_2)^2 + (y-b_2)^2.$ When the squares are expanded, the $x^2$ and $y^2$ terms appear on both sides with coefficient $+1$ and cancel. The result is therefore a polynomial of the first degree in $x,y$, that is the equation of a line.

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Topics: Loci
Concepts: Perpendicular bisector of a segment · Distance between points · Equation of the line · Cartesian plane · Line
Skills: Proving · Analytic geometry