Proof for Laws of Reflection Using Huygens' Principle

Using Huygens' principle, the law of reflection can be derived directly from wave geometry. A plane wavefront $AB$, perpendicular to incident rays $L,M$, strikes a plane mirror $XY$ obliquely, so point $A$ reaches the mirror before point $B$ does. By the time $B$ finally reaches the mirror (at $B'$), $A$'s own secondary wavelet has already expanded to $A'$, with $AA'=BB'$ (since both travel the same speed for the same elapsed time in the same medium) -- the new reflected wavefront is the tangent line $A'B'$. The two right triangles $\triangle ABB'$ and $\triangle B'A'A$ this construction creates share a common hypotenuse $AB'$ and equal legs $AA'=BB'$, making them congruent; this congruency forces the marked angle of incidence to equal the marked angle of reflection, proving $i=r …