Hey, everyone! For those of you who don't know me (most of you), I'm Sotiri and I'm a freshman at the University of Chicago. I am a member of the TorqueGroup (aka Torque Team) hashtag and hope to bring everyone some interesting or thought-provoking topics in the near future from my studies in math both in and out of school :)
With that in mind, I think a good topic for my first post is "proofs using complex numbers." Knowing how to do said proofs is very important, as very often complex numbers can be used to greatly simplify problems involving other topics, especially geometry and trigonometry. We should all take a moment to thank Euler for his marvelous discovery of the formula . It truly is amazing.
Anyways, I'll demonstrate how to use the concept of complex numbers to prove one of the most famous theorems in all of mathematics: Heron's Formula! I originally found this proof in this paper on Art of Problem Solving, but I do not know who originally came up with it. Regardless, it is one of my favorite proofs, and I will reconstruct it here with plentiful explanation.
In the image above, is the incenter of . This provides justification as to why the three segments emanating from and perpendicular to the sides are congruent and have length (they are radii of the incircle of ). Also, we can see that, as labeled, the central angles come in congruent pairs, since the incenter is the intersection of the angle bisectors of the triangle. Finally, since tangents from a point to a given circle are congruent, we see that the sub-segments on the sides of the triangle also come in congruent pairs.
We let . Clearly, the semiperimeter and . Now we introduce complex numbers into the problem by setting . This is just a simple application of Euler's formula after defining a proper set of coordinates. For example:
We multiply everything together to get
so that the imaginary part of the product is . However, we note that and so . It follows that .
We remind ourselves that and , and so . But we know that we can express the area as with the exact same and as we've already been using (it is well-known that the area is the product of the inradius and the semiperimeter), and so we finally get , as desired.
I hope the proof itself was clear, interesting, and shed some light on how powerful complex numbers can be in geometric proofs. It should be noted that we did almost no algebra at all other than the simple expansion of the product. The problem took a bit of time to set up, but after that, everything was very straightforward.
If you guys are interested in seeing more, I'd be happy to work through more complex-number proofs, especially of old IMO problems. Otherwise, please do let me know if there are other topics I should write about. For now, you all should try to solve this problem using complex numbers:
The above figure shows an arbitrary quadrilateral with a square drawn on each of its sides. Show that and that the two segments are perpendicular.