Heron's principleHeronsPrinciple2009-02-13 13:25:552009-02-14 15:48:00Theoremreflection% this is the default PlanetPhysics preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here\PMlinkescapeword{side} \PMlinkescapeword{sides}
\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 {\em Heron's principle}, concerning the reflection of light, is a special case of {\em Fermat's principle} in optics.\\
{\em 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 of $l$ set 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 line of $l$ set through $C$ intersects $l$.
Justification:\, From two pairs of similar right triangles we get the proportion equations
$$AP:CX \;=\; PQ:XQ, \quad BQ:CX \;=\; PQ:PX,$$
which imply the equation
$$AP:PX \;=\; BQ:XQ.$$
From this we can infer that also
$$\Delta AXP \sim \Delta BXQ.$$
Thus the corresponding angles $AXP$ and $BXQ$ are equal.\\
. . .