Featured Member - Xuming Liang

We will be featuring different members of the Brilliant community, so that you can get to know them better. For the twelfth issue, we are featuring Xuming Liang, who recently developed a theorem and published a paper in the International Journal of Geometry

Tell us more about yourself

I do not consider myself a genius (even though others try to label me), as there are many areas that I struggle in. For example, since I came to America 8 years ago, language has never been a strong suit of mine. You can expect grammatical errors and unconventional interpretations in what I write.
I am considered a late bloomer in mathematics. I did not consciously participate in math competitions till I was in ninth grade, and I was average before that. When I joined the math club, I could barely solve any of the AMC problems, which sparked my desire to learn. I found myself immersed in the math competition culture, and periodically visited AoPS, self-learnt from downloaded pdfs, and solved a lot of problems on Brilliant. Through this process, I grew to love problem solving, and was further motivated to learn math.

Tell us something that others do not know about you.

I'm an atypical asian nerd, because I'm into electronic dance music.
I am interested in Psychology and Sociology, particularly the system of thoughts inside our head and how it relates to our behaviors.
I can spin pens really well. In fact, I used to participate in online pen-spinning competitions. (Before I got obsessed with math of course.)

Describe the Liang-Zelich Theorem
Liang-Zelich theorem describes an invariant property among orthologic triangles (a pair of triangles in which the perpendiculars from the vertices of the first triangle to the sides of the second triangle are concurrent). It states that if $P$ satisfies the property that $PP’$ interests the Euler line of $ABC$ at a given ratio relative to the circumcenter $O$ and orthocenter $H$, where $P’$ is the isogonal conjugate of $P$ wrt $ABC$, then $P$ also satisfies the same property with respect to its pedal triangle of $ABC$.

What motivated the theorem?
Well, the “given ratio” in the statement hints that it is a generalization of a particular ratio. Indeed! Prior to discovering this theorem, Ivan and I spent a month of two on a problem on AoPS without any synthetic solution at the time. When we solved it, our solution was long and rich in techniques. Ivan and I planned on writing an article about it, but what hindered us was the lack of cohesiveness in the properties we discussed. I also doubted that the an article was worth writing. Here was the problem we solved:

Let $O,I$ denote the circumcenter and incenter of $ABC$. Prove that the nine-point center of the cevian triangle of $I$ lies on $OI$.

Shortly after we solved it, I independently found a much simpler answer, which is posted on my AoPS blog. My solution prompted me to investigate the nine-point center configurations, and it led to the discovery of the following generalization:

If $P$ satisfies that $PP’$ passed through the nine-point center of $ABC$ where $P,P’$ are isogonal conjugates of $ABC$, then $P$ satisfies the same property with respect to the pedal triangle of $P$.

This is a much harder problem than the one above it, and before Ivan and I could settle down to consider this one, I bravely tested if it could be further generalized. To my utter surprise, the property generalized nicely as the nine-point center can be replaced by any arbitrary point whose position is determined relative to $O,H$ on the Euler Line, i.e we replace nine-point center in the proposition above with $T$ where $TH/TO$ is the given constant.

How did you prove the theorem?
At first, we did not have any ideas on how to prove it. We tried to reframe the problem and produced many equivalences, none of which seemed approachable. The heart of the problem is to show that two ratios are equal, and it was clear that we needed to find some medium to bridge them together. The main obstacle was working with ratios involving central points $O,H,P,P’$, which by themselves are immobile and difficult to calculate. In the midst of our clueless inquiry, Ivan came to the rescue with the introduction of a very crucial property that related the ratios with perspective triangles. For once, we have a way to express the seemingly unmovable ratio $TH/TO$ in terms of perspective triangles; we had found our appropriate medium!

Seeing that all we needed was to prove this essential property, Ivan and I became more motivated. After a month of back and forth exchange of ideas, we finally slain the beast! Ivan later informed me that he found the crucial property by noticing similar properties occurring for several well known cubics from this website.

What are some applications of the theorem?
Using the theorem, we were able to prove an open problem!

Let $ABC$ be a triangle, and $O$ be its circumcenter. The perpendicular bisectors of $BC$ and $AO$ intersect at $O_a$. Define $O_b, O_c$ similarly. What point represents the intersection of the Euler lines of $O_a O_b O_c$ and $ABC$?

We were also able to provide numerous short proofs to the properties of the McCay cubic, Thomson cubic and Darboux cubic. Several of the old proofs were extremely complicated, but our insights provided us with the right descriptions to demonstrate these properties.

We consider this theorem as a fundamental result in Geometry, as it allows us to easily find new properties of triangle centers in the pedal triangle. We suspect that this theorem could be generalized projectively, to isopivotal cubics invariant under isotomic conjugation.

What do you want to accomplish?
Academically, a short term goal of mine right now is to seize my final opportunity to do well in the USAMO. It has been my dream to participate in the IMO since nineth grade; I cannot believe this is my last year already.
I really enjoy doing research, so I cannot wait to have similar opportunities in the future. Something specific would be contributing to the understanding of a difficult math concept, as well as collaborating with like-minded others.
On the personal side, I hope to achieve financial stability, help those in need by using what I know, and live a happy life.

What do you wish for Brilliant?
Brilliant is no doubt a great site for those who love math, science, and problem solving. The diverse community very welcoming. Brilliant provides me with a platform to share what I've learnt, and makes my passion for mathematics more meaningful. My self-motivation for learning topics of interest, can be shared with others and we all benefit. I wish that more people in the future would contribute back to Brilliant in some way or form. I started out as a beginner on this site and now I want to give back.