Corner Reflector Proof

You Really Should Figure This Out Yourself

But I'll assume you've tried. The black lines in the upper right corner are the two mirrors, and the red lines are the light beam coming in, reflecting twice and leaving. We want to prove that the incoming and outgoing light, the two red lines that look parallel, actually are parallel.

Here's one way, that only requires one magic line to be added to the basic figure: the short black horizontal line is contructed at right angles to the mirror on the right, where the red light beam reflects. We've also extended one of the red lines through the mirror: this is geometry, not physics. Five angles are labeled, A through E. We're going to show, step by step, that they're all equal. If you mouseover the pointing hands below, that step is highlit.

First, angle A equals angle B, because that's the rule for reflection. Next, angle C equals angle D, for the same reason.

Angle E equals angle B because they're formed by two crossing lines. (E plus the big unlabeled angle is a straight line angle of 180 degrees, and so is B plus the same big angle equal to the other straight line angle) Hey, this is easy!

Now the trickier one, E = C. You need to remember some geometry for this one. In particular, when you have two parallel lines and a third line crossing them, the angles formed are equal. And the reverse: if the angles are equal, the lines are parallel. The two horizontal black lines in the big figure are parallel, since they form equal angles with the vertical black line. (Both are right angles, since we defined them that way...this is where the fact that the mirrors are at right angles to each other is used)

Since the two horizontal black lines are parallel, the slanting red line forms equal angles with them, and angle E equals angle C.

So all the angles are equal, A = B = C = D = E.

Now the final trick! the incoming and outgoing red lines are parallel because they make equal angles with the other short red line: the angle with the left red line is A+E, and the angle with the right red line is C+D, and those are equal since all the labeled angles are equal.

We didn't use any special property of the angle A, so the incoming and outgoing light is parallel at any value of angle A. We did use the fact that the mirrors are perpendicular, so this proof is valid only for a 90 degree corner reflector.

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