A brilliant way to prove the exterior angle sum of a polygon!
Here's a triangle, and we want the sum of its exterior angles. The exterior angle sum is defined as "the sum of the supplementary angles of every interior angle of a polygon" — in this case, ∠DAC+∠FCB+∠EBA. Of course, this generally refers to convex polygons.
Exterior angle sum of a triangle (1)
Clearly this isn't much of a challenge — the answer is 360 degrees, and there are many ways to get there: computing it directly from the interior-angle-sum formula, imagining a full rotation, or even just measuring it yourself. But I think the most brilliant method is undoubtedly the one below. more
Looking at the picture above, it's hard to conclude that the exterior angles sum to 360 degrees — but that's only because we're standing too close to it, so "we cannot see the true face of Mount Lu, simply because we ourselves are on the mountain." Let's thicken the lines a bit (so we don't lose sight of the segments), and then step back and look at it from farther away.
Exterior angle sum of a triangle (2)
Still can't see the truth? Then let's thicken the lines further, and step back even farther.
Exterior angle sum of a triangle (3)
Keep repeating this... Finally, in for a penny, in for a pound — let's just go stand infinitely far away, at a distance where even the triangle itself becomes invisible. We get:
Exterior angle sum of a triangle (4)
What we know about magnifying glasses tells us that a magnifying glass can change lengths, areas, and volumes, but it cannot change the size of an angle. So no matter how far away we stand, the scenery we see shrinks, but the angles don't change. Now the triangle has shrunk down to a single point — what do we see? Three angles! Their sizes are exactly those three exterior angles, and together they fit around a full circle! Clearly, their sum is 360 degrees! So the exterior angle sum of a triangle is 360 degrees!
Brilliant! This is a limiting argument, and also a display of cleverness and playfulness: I was asked to find the exterior angle sum of a triangle, but I was never told how large that triangle had to be — so I might as well find the exterior angle sum of an extremely tiny triangle, which is far easier...
The proof for a general polygon can be constructed analogously... It's this kind of thinking that makes mathematics interesting and beautiful.
(This idea comes from the "Science Squirrels Club")
Translated automatically with claude-sonnet-5; all equations are reproduced verbatim from the source. Copyright remains with the original author.