Heron's principleHeronsPrinciple2009-02-13 13:25:552026-09-05 23:09:49Theoremcommenting out pictures until we can upload themreflection% this is the default PlanetMath preamble. as your knowledge
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\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}\textbf{Theorem.}
Let $A$ and $B$ be two points and $l$ a line of the Euclidean plane.
If $X$ is a point of $l$ such that the sum $AX+XB$ is the least possible,
then the lines $AX$ and $BX$ form equal angles with the line $l$.
This \emph{Heron's principle}, concerning the reflection of light, is a
special case of \emph{Fermat's principle} in optics.
\emph{Proof.}
If $A$ and $B$ are on different sides of $l$, then $X$ must be on the line
$AB$, and the assertion is trivial since the vertical angles are equal.
Thus, let the points $A$ and $B$ be on the same side of $l$.
Denote by $P$ and $Q$ the points of the line $l$ where the normals to $l$
through $A$ and $B$ intersect $l$, respectively.
Let $C$ be the intersection point of the lines $AQ$ and $BP$.
Then $X$ is the point of $l$ where the normal to $l$ through $C$ intersects
$l$.
\begin{center}
\includegraphics{HeronsPrinciple_construction.png}
\vspace{0.5em}
\textbf{Figure.}
Construction used in the proof of Heron's principle.
\end{center}
Justification:
From two pairs of similar right triangles we get the proportion equations
\[
AP:CX = PQ:XQ,
\qquad
BQ:CX = PQ:PX,
\]
which imply
\[
AP:PX = BQ:XQ.
\]
From this we can infer that
\[
\triangle AXP \sim \triangle BXQ.
\]
Thus the corresponding angles $AXP$ and $BXQ$ are equal.
\begin{center}
\includegraphics{HeronsPrinciple_reflection.png}
\vspace{0.5em}
\textbf{Figure.}
Reflection construction showing why the path through $X$ is shortest.
\end{center}
It remains to show that the route $AXB$ is the shortest.
If $X_1$ is another point of the line $l$, then
$AX_1=A'X_1$, and therefore
\[
AX_1B
=
A'X_1B
=
A'X_1+X_1B
\geq
A'B
=
A'XB
=
AXB.
\]
\begin{thebibliography}{8}
\bibitem{th}
Tero Harju,
\emph{Geometria. Lyhyt kurssi}.
Matematiikan laitos. Turun yliopisto, Turku (2007).
\end{thebibliography}