Enter point coordinates and line equation to compute the reflected point
Point P₀
x₀
y₀
Line Ax + By + C = 0
x +y += 0
Result
Point P₀(, ) reflected about the line is
Step-by-Step Derivation
Reflection Formula
x = x₀ − 2A(Ax₀+By₀+C)/(A²+B²) y = y₀ − 2B(Ax₀+By₀+C)/(A²+B²)
Derived from two conditions: 1) PP\' is perpendicular to line L; 2) The midpoint of PP\' lies on line L.
⚠Note: A and B cannot both be 0. When the point lies exactly on the line, the reflected point is the point itself.
What Is Point Symmetry About a Line?
Point reflection about a line is an important transformation in plane geometry and analytic geometry. If point P\' is the reflection of point P about line L, then L is the perpendicular bisector of line segment PP\'.
Perpendicular Condition
PP\' is perpendicular to L. Slope of L is −A/B (B≠0), slope of PP\' is B/A (A≠0). Product = −1.
Midpoint Condition
Midpoint M((x₀+x)/2, (y₀+y)/2) lies on L: A·(x₀+x)/2 + B·(y₀+y)/2 + C = 0.
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