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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  .
Justification: From two pairs of similar right triangles we get the proportion equations
which imply the equation
From this we can infer that also
Thus the corresponding angles  and  are equal.
We still state that the route  is the shortest. If  is another point of the line  , then
 , and thus we obtain
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- TERO HARJU: Geometria. Lyhyt kurssi. Matematiikan laitos. Turun yliopisto, Turku (2007).
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