The figure below shows a triangle, and we want to find the sum of its exterior angles. The sum of exterior angles is defined as “the sum of the supplementary angles of each interior angle of the polygon,” such as \angle DAC + \angle FCB + \angle EBA here. Of course, this generally refers to convex polygons.
Obviously, this is not a major problem; the answer is 360 degrees. There are many methods: calculating directly using the interior angle sum formula, imagining rotating one full turn, or even measuring it yourself. But I think the most brilliant method is undoubtedly the one below.
Looking at the figure above, it is difficult to conclude that the sum of the exterior angles is 360 degrees. However, that is only because we are too close to it, so “we cannot see the true face of the mountain because we are standing on it.” Let us thicken the lines a little bit (so that we don’t lose sight of the segments) and then observe it from further away.
Still can’t see the truth? Then let’s thicken them again and move even further away.
Repeat this... Finally, “once we start, we might as well go all the way,” and simply stand at an infinite distance where even the triangle cannot be seen clearly, obtaining:
Knowledge of using a magnifying glass tells us that a magnifying glass can change length, area, and volume, but it cannot change the size of an angle. Therefore, no matter how far away we stand, the objects we see shrink, but the angles do not change. Now that the triangle has shrunk to a point, what do we get? Three angles! Their sizes are exactly those three exterior angles, and they combine to form a full circle! Obviously, their sum is 360 degrees! Therefore, the sum of the exterior angles of a triangle is 360 degrees!
Wonderful! This is a type of limit thinking, and also a manifestation of cleverness and playfulness: He asked me to find the sum of the exterior angles of a triangle, but he didn’t say how large the triangle should be. I’ll just find the sum of the exterior angles of a very, very small triangle, which is much easier...
General polygons can also be proven by analogy in this way... It makes mathematics interesting and beautiful.
(This idea comes from “Songshuhui”)
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