Update on Exploring Euclidean Geometry.

Well, here goes nothing.

Preface and other Minutia

I’ve redone the Preface for what’s probably the fifth or sixth time. I am quite happy with it this time around and will worry about it later. I’ll pretend it’s good because I don’t want to stare at “Don’t do the preface because it’ll suck and take time away from actual book-writing.” Not anything major, but nice to have done since Author’s Note/Dedication has been completed already.

Also, I recently remembered to take off the “anyone with link can view,” and am not going to show the book until it is ready to release (except for a couple of people who 1) I know well and 2) have specifically requested to read it). I will ask on AoPS for proofreaders, though I will have to remember to watermark the pdfs. (I forgot to do so when sending to a friend – however, since they are a friend, I trust them not to leak.)

Isogonal and Isotomic Conjugates

I’ve scoured around the web and found some classic isogonal theorems. Evan’s post was helpful for the harder theorems, but easier theorems were left as exercises. I also included a particularly nice perpendicularity theorem I stumbled upon; I forget whether it is in my blog or not as of the moment.

Right now, the isogonal section isn’t even finished. I’m planning to have a nice divide of

  1. Isogonal Theory
  2. Isogonal Problems
  3. Isotomic Theory
  4. Isotomic Problems

Right now, I need a couple of things to bolster the Isogonal Problems and I need to add Symmedians to Isogonal Theory. Isotomic, I feel, will mostly have to be developed “ground-up” – it’s not really the most used or well known configuration.

Note: I began the draft at January 8th, 11:21 AM, which is a Tuesday. Considering that it was probably during a “work period” (which is stupid – just extend lunch! But that’ll wait until a later post), I feel like I should’ve started earlier. It’s been two weeks and I’m not halfway done. Damn.


Taking a look at the AoPS Volume 2 book, their attitude towards projection (and Inversion/Homothety, both of which I have covered) is “these are cool to include, but oops I’m getting lazy so I’ll just add cursory information.” In my opinion, Homothety is the best place to include Nine-Point Circle and Euler Line (two important things oddly missing from Volume 2).

This is a pity, as I’ve frequently referred to AoPS V2 for stuff such as Trig. (Although some of their proofs are outright weird, and it doesn’t help they don’t have a Trig formula sheet. They list formulas or skeletons of formulas and their proofs, which is analogous to what I did. However, they’ve mixed together stuff that I think belongs on three chapters, though I split into four for some strange reason.)

Getting back on topic, I’ll probably use AoPS V2 for the beginner’s stuff about projective, then I’ll have to rely on murky prior knowledge. Just so I don’t forget, you can project any conic to another conic. (This is useful especially if you don’t want to deal with the weird shenanigans of something, say, inscribed in a parabola and would rather look at a circle instead. Isn’t that right, Pascal?)

However, when I look at AoPS V2, all I see are projections used for ellipses, when a dilation could do the same. This makes me wonder whether projections are truly a useful thing of their own, or if they are just used to hand-wave most conics and just to say “consider a circle instead.”

Note to self: use http://alexanderrem.weebly.com/uploads/7/2/5/6/72566533/projectivegeometry.pdf for help on projective.

It also seems projective is mostly harmonic bundles.

Release Date

Isogonal/Isotomic + Projective + Conics will bump the release date to around June, if I’m optimistic. And these three are the only I’ve done cursory research into – who knows how 3D Analytic (aka 3D Vectors) will turn out? Other than “Definitions and the Basics” for 3D Geo, none of these chapters seem like “freebies.”

I’ll be honest and say each major chapter might take a month at least, and the minor ones added together could take anywhere from 2 weeks to 6 weeks. Considering we definitely have at least 3 major chapters and probably 4 (oh no I might have to actually learn Analytic again by using my chapter on Vectors), and the tidbit on Geometric Inequalities might be hard (and annoying).


Some Isogonal Conjugates Theorems.

Recently I stumbled upon Evan’s Isogonal Conjugates post on WordPress, and I have a couple of theorems that I feel are worth adding along to his post to my own blog.

1. Let Q be the isogonal conjugate of P with respect to ABC. Let AP,AQ intersect BC at X,Y, respectively. Then (BX*BY)/(CX*CY)=c^2/b^2.

Proof: Use areas (I haven’t actually done this) or Sine Law (I know this proof is slick).

2. Consider ABC and point P with isogonal conjugate Q, and let P have pedal triangle DEF. Then AQ⟂EF.

Proof: Use right triangles and the trivial isogonal angle equality theorem.

You can use this to provide a “brain-dead” proof for Theorem 3 (Pedal Circles).

A final note: Pascal’s Theorem is true for any conic and an inscribed hexagon, though the proof is pretty much identical. This is because you can project a conic to an ellipse, and project/dilate it to a circle, where the proof proceeds identically!

Evan’s post is here: https://usamo.wordpress.com/2014/11/30/three-properties-of-isogonal-conjugates/

Website Updates

I’ve updated the website so that it contains a sample of Exploring Euclidean Geometry.

The section included is F4 – Inversion. Take a look here.

Projected release: Those of you who read my last post know this is mid-April, but I am not exactly sure when it will be.

Also, the entire website will be rebranded as “EEG+other stuff.” This process starts today.

The New Year

With the New Year that has just arrived, I have an opportunity to revise the dates for documents for the next four months when I inevitably forget it is 2019. While I’m at it, I might want to revise my life as well.

  • Exploring Euclidean Geometry – Those who know me in person have already heard about this project. For those of you who don’t know, EEG is a geometry book leaned more towards those who are interested in learning something new in geometry instead of competition math. I have a couple of things I need to do for the book.
  1. First and foremost, finish all of the theory. Write a couple of problems for each section.
  2. Grind out geometry problems in general, adding them to whichever section is most fitting.
  3. When I feel there are enough problems to release this as a book, I will write a preface. (Note to self: Don’t go too overboard. Last time, I wrote 4 or 5 pages worth of introductions… during a standardized test.)
  4. Branding, website preparations, etc. (In particular, the home page of the website will advertise the book, so I actually have a use for my home page. In general, homepages are useless – but I digress.) Instead of having this website be known as “Dennis Chen’s website + book,” I will have this be known as “Book (also see some handouts).”

General notes on the geometry book:

Three sections on transformations to write. One currently written section (homothety) desperately needs more problems. Three sections on 3D Geo to write (I really do NOT want to do Analytic or Conics, because I’ve done enough of them in 2D Geo). 4 more miscellaneous (and short) sections to write, and I will probably pad the end of the book with some parting shots. 3+3+4=10, so 10 sections left to write.

The last section took me 13 days to write, but I think I can do better – churning out a section every 3 days should be possible for the shorter ones, because I’ve been busy with USAMTS and a “challenge.” Projective and 3D Analytic will probably take a week each. 3*8+2*7=38, so 38 more days of work and the book should be out. Since I have no school, I’ll subtract a couple of days and estimate 35. The problems will probably take me 2 months at most (I already have a couple in mind for some sections), preface/branding will take a week. I hope I can release the book by April.

(Of course, unless I have more content to add. Though I’m already writing stuff like Projective and by the time I’m done, I’ll probably be tired of geometry.)

  • Exercise Book to accompany EEG – as self-explanatory as it gets. I have a couple of nice problems in mind (USAMTS #5 from Round 2 in particular) to put on there, and I have some problems which I’m pretty sure I’ll forget on there. However, this isn’t very high priority with the actual book (with problems from yours truly) and the MPP Summer Camp (see below) coming up, which I care about a little more. Estimated release date: Probably 2020 at earliest.


  • MPP Summer Camp – This one really is in the works. I have to hope I’ll be done with the Geometry book (minus branding) by early March. Currently planned topics: Root Analysis, Factoring, Complex Numbers. I wanted to teach some Algebra (MPP was mostly about Geometry to this point), so that’s what’s planned. However, none of the handouts are written – though doing some old AIME’s has gotten the Root Analysis part pretty much done. I just have to write stuff.


  • Smash Ultimate – Get good at the game. I realized I have never blogged about this, ever. I gave up on my old main Kirby (who I religiously stuck by in Smash 4), and picked up some better characters. This is my character lineup:
  1. Pikachu – NEW MAIN NEW MAIN NEW MAIN! I used him in Smash 4 and boy is he much faster, stronger, and F-Smash is safe on shield! Thunder is ridiculous (which Pichu will bring to new heights), the new N-air lends so well into combos, and D-Tilt is plain stupid. There’s a reason some top players call him #1.
  2. Pichu – Despite Pikachu being my proclaimed main, I probably use Pichu more. Why? Because of his N-air and Up-air! I can run around the whole stage and spam those aerials to control the flow of the game. Yes – he hurts himself, but his best moves (N-air, U-air, U-tilt) don’t hurt himself. Also, the 1 percent he deals on Thunder is well worth the reduced lag. Pichu can still reasonably control stage with Thunderjolt, you can probably just charge Skull Bash and it’ll work, and F-Tilt, a move that you can spam, is a kill tool. The only complaints I have are that his F-Smash has awful range, and his Up-B does too much self damage and has no hitbox.
  3. Mewtwo – Two words: Forward. Aerial. (Also, Shadow Ball has so much stage control.) I was secretly hoping Mewtwo’s float from PM would come, but I’ll take an OP F-air over nothing any day.
  4. Meta Knight – His Up-B felt nice initially, but it’s too hard to connect. His ‘Nado being a kill tool though is welcome. The Rufio from Brawl returns (kind of) as he can kill with his ‘Nado (through the side, but details don’t matter). However, he’d go into my Pockets category.

So now I need to practice the Dash-Attack cancel tech. Goodbye. (For those of you who don’t know what I’m referring to, My Smash Corner made a video about it.)

(Notes to self: Reference AoPS V2 for Conics. V2 is a good start for Projective but research will probably need to be done on my own. Analytic = Vectors [mostly] as 3D neq 4D. Remember La Hire’s exists.)


After not posting something for long, I’ll post something a little bit underwhelming. The inactivity wasn’t because I had something really good right now (the only thing I have on the back burner is satire), but it’s because I’ve been busy with school work. With winter break, I’ve gotten more time to work on the things I care about (i.e. NOT school).  I hope you find this interesting, despite this post having no substance.

Remembering that all your students exist

I have 6 wonderful students and one not so wonderful student (screw you, Steven). When I need to list them all or take mental attendance, I have to get out the roster. I realize that I have 5 or 6 people listed and blankly wonder who I forgot, despite all 7 of them being in front of me. I pull up the roster and look for 10 minutes before I realize who is missing, and it seems very obvious to me after I find it.

Then we have teachers whose classes have 30 people a period, and have around 90 to 150 people in total. They can remember who exists without batting an eye; often attendance is done without looking at the seating chart and just by memory of seating arrangement. It truly amazes me how teachers can do that, when I can’t remember 7 people who I converse with often outside of class.

And we don’t understand why seating changes aren’t more frequent.

(I am very sorry, class, but it really is hard to list all of you guys. This probably speaks volumes about my teaching method, which is to make the lecture not tailored to anyone at all, so that the kids will think I am tailoring it to everyone.)

Trying to make problems

Boy, this is going to piss my class off. As if I didn’t already last time.

Whenever I make a conscious effort to make problems (i.e. flip through pages of study guides and textbooks to find an idea I can use), they usually turn out to be crap. (Around 5-10% of my problems are going to make it on, say, the final version of our entrance exam.) In contrast, my shower problems (usually brought on because of a combination of fascination for some idea, stupid flavortext ideas, and guilt that nothing I made in the last hour was remotely salvageable), I have a success rate of around 30-40%, and that is a very stingy estimate. At worst this is an improvement of x3, which I have no idea why.

Obvious Observations

Recently I was given the trivial line $$\angle HBC=90^{\circ}-\angle C.$$ The reasoning took me 10 minutes to realize. For those of you as lazy as me, this is because extending HB makes a right triangle. Oops.

I’m sure I have other idiocies, like $$\angle AOB=2\angle ACB,$$ but I don’t remember any of them as of the moment.


Disclaimer: I usually claim to be knowledgeable in what I blog about, but this time I do not claim to be. If you’re an expert, feel free to correct me, since I’ll probably be wrong most of the time.

Generally, when I write a handout on something, I’ll have two things: theorems, and problems. Theorems used were usually proved very long ago, and problems can easily be sourced as “2018/AMC 10A/25,” though (unfortunately) people don’t do the best job at citing and leave it as “2018 AMC 10,” which makes it take a while to find the problem.

Let’s take a look at what MLA has to say about citing your sources.

So, MLA, there’s this very helpful book for aspiring mathematicians. It is called, “the Art of Problem Solving: Volume 2.” Those of you who don’t know this are probably going on google.com and searching it up, in which case it would be a good citation, since I made it easy to find.

Take for example, this imaginary quote from an imaginary chapter of my imaginary book.

A History of Logarithms

Chapter 1 – What is a Logarithm?

To understand the history of logarithms and their uses, we must first understand what exactly a logarithm is. For this task, I turn to Chapter 1 of “the Art of Problem Solving: Volume 2” for the definition of the logarithm and the six most important properties.

[insert definition]

[insert properties]

Now let’s let the MLA do this for us!

A History of Logarithms

Chapter 1 – What is a Logarithm?

To understand the history of logarithms and their uses, we must first understand what exactly a logarithm is.

[insert definition]

[insert properties]

Then, you get to flip to the end of the book, find the bibliography (this is much more annoying the further in you get), and it will say this following:

Works Cited

Rusczyk, Richard, and Sandor Lehoczky. The Art of Problem Solving. AoPS Inc., 2013.

Insert source here.

Hanging indents suck.

Here’s my initial reaction to this:

When did you use this source? Which AoPS book is this? What counts as a citation? What section did you use? What is wrong with you?

After I calm down, my thought-out and reasonable response would be this: “You can go screw yourself.”

(Note: The original draft of this post had an endnote, and a rant on why footnotes are far superior. Then I remembered that many people will simply put “Bibliography” and have no reasonable way aside from guessing to know where each reference fits in, making it even more annoying.)

How to Fix MLA

I’m not an English/History teacher or professor. If my peers did this, I would be extremely annoyed, but they don’t. Ideally, English teachers would have other English teachers to tell them that they need to stop making their citations so lengthy, effort-requiring, yet worthless. This should mean that I don’t care about MLA citations, since it’s not my place and they don’t affect me.

However, due to the dreadful government institution that serves barely-edible food, has people stand in an orderly grid-ish fashion to take rollcall, has iron gates as a security feature, has officers that patrol the campus, is filled with zero-tolerance policies, and has a schedule strictly to the minute, this becomes my business. (And for those of you wondering whether this is school or prison, this is school. Prison inmates don’t have to write English essays, which may be the only difference.)

The true fault does not lie with the MLA. The fault lies with English and History teachers around the country. If they’d let me write my papers the way I write them, while actually demanding a satisfactory result (i.e. don’t give me 100% for sucking up in the entire essay), I wouldn’t complain about the actions of a far-off organization. When they make me adhere to the standards of a far-off organization, I expose the sheer stupidity of said standards.

TL;DR: Stop making kids do MLA citations!