theorem. Let and be two points and a line of the Euclidean plane. If is a point of such that the sum is the least possible, then the lines and form equal angles with the line .
This Heron's principle, concerning the reflection
of light, is a special case of Fermat's principle in optics.
Proof. If and are on different sides of , then must be on the line , and the assertion is trivial since the vertical angles are equal. Thus, let the points and be on the same side of . Denote by and the points of the line where the normals of set through and intersect , respectively. Let be the intersection point of the lines and . Then, is the point of where the normal line of set through intersects .
which imply the equation
From this we can infer that also
Thus the corresponding angles and are equal.
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