Senior Michael Scheer recently developed an original proof of Feuerbach’s Theorem, which is explained in his soon-to-be-published paper titled “A Simple Vector Proof of Feuerbuch’s Theorem.” The paper will be published in an upcoming issue of the mathematics journal Forum Geometricorum. Forum Geometricorum is a free, annual math journal which was first published in 2001 by the Department of Mathematical Sciences at Florida Atlantic University.
Feuerbach’s Theorem, first proved in 1822 by Karl Feuerbach, states that a nine-point circle is tangent to the incircles and excircles of a scalene or isosceles triangle. An incircle is a circle inscribed in a polygon whereas an excircle is a circle that is tangent to a side of the triangle as well as the lines containing the segments that make up the other two sides of the triangle. A nine-point circle is a circle that passes through the midpoints of the sides of a triangle, the foot of each altitude, and the midpoints of the line segments extending from the triangle’s vertices to its orthocenter, the point where all three altitudes meet.
“Every triangle has four circles that are tangent to all three sides of the triangle [the incircle and the three excircles]. Feuerbach’s theorem tells us that there is a circle that is tangent to all four of these circles, and, furthermore, this circle contains nine important points on the triangle. This really is a crazy coincidence,” Scheer said in an e-mail interview.
Scheer first stumbled upon Feuerbach’s theorem during his freshman year Geometry class. “I had [former mathematics teacher Joseph] Stern for Geometry second term, and in his class we covered the nine-point circle. And after we finished discussing the proof of its existence, he gave us a choice: move on with the curriculum and learn about coordinate geometry, or prove Feuerbach’s Theorem,” Scheer said in an e-mail interview. Though the class originally chose to learn the proof of the theorem, which Stern had said might take two weeks to explain, they reverted back to the curriculum in two days. Scheer’s curiosity, however, was not satisfied. “I was still really interested in [the theorem] and wanted to know why it was true,” he said.
On his own time, Scheer read through different proofs that had been proposed by mathematicians in the past. However, most of them were very complicated and involved higher-level mathematics, making them difficult for a high school mathematician to fully comprehend. During his sophomore year, Scheer started playing around with different mathematical topics, specifically applications of vector algebra in geometry. In his attempts to do so, he discovered “a very nice formula giving the distance between two points as a function of things that are easier to compute,” Scheer said in an e-mail interview. “After finding this formula, I realized that I might be able to prove Feuerbach’s Theorem with it, because Feuerbach’s Theorem can be recast entirely in terms of the distances between various points related to a triangle,” Scheer said. “So I tried this out, and lo and behold, half an hour later, I had a proof of Feuerbach’s theorem.”
Excited about his accomplishment, Scheer e-mailed Stern, his freshman Geometry teacher. Stern suggested that Scheer publish his paper in the mathematics journal Forum Geometricorum, which Stern had previously been published in. The two began to collaborate, editing the paper in a series of e-mails. They formatted the paper, adding images that made it easier to follow and understand, before Scheer sent the paper to the publisher of Forum Geometricorum. Math teacher Asvin Jaishankar provided his expertise in ensuring that there were no grammatical or other flaws that interfered with the paper’s flow. It was forwarded to a referee who checked the originality of the proof and ensured that it was not flawed. The referee’s positive response prompted the publisher to accept the paper. Currently, Scheer and Stern are completing the editing process.
“What makes a mathematical paper striking, much like papers written in any academic field, is that it is well-organized and the explanations are clear and concise. [Scheer] begins by not only stating the theorem, but also how he intends to go about proving it,” Jaishankar said in an e-mail interview. Jaishankar also praised the clarity of Michael’s paper, which does not skip any computational steps in the presumption that the readers understand the level of mathematics he used. “This is not an easy theorem to prove, and if you don’t include the work, you will lose much of your reading audience before they even get to the crux of the paper,” Jaishankar said.
Stern could not be reached for comment.
The significance of Scheer’s proof arises from the fact that it allows people who are not mathematical scholars to understand the proof of Feuerbach’s Theorem. His proof does not use difficult mathematical concepts, but is constructed out of vector principals. “Its relative simplicity and elementary nature are its most unique aspects,” Scheer said in an e-mail interview.
The imminent publishing of Scheer’s paper is no doubt a milestone in his exploration of mathematics. “I’m really happy that I can finally start giving back to the mathematical community, not just consuming mathematics,” he said in an e-mail interview. Scheer hopes that this paper will be the start of his career in math, if he eventually decides to pursue a job in that field.
“Honestly, I don’t think he has scratched the surface of what he can achieve, and I look forward to seeing a lot more from him in the future,” Jaishankar said.
Indeed, Scheer’s enthusiasm for mathematics will probably keep him at work. “Doing original mathematics is like exploring a new place. Nobody knows how it will work yet and really anybody can do it,” Scheer said.