In this blog I’ve been posting controversial opinions; here is one that (hopefully) is more popular.


Before talking about a grudge, we should define it.

Let’s consult the dictionary!

As a noun:

a persistent feeling of ill will or resentment resulting from a past insult or injury.
Let’s jump down the rabbit-hole a little more; what does resentment mean?
bitter indignation at having been treated unfairly.
Again, another big word. Let’s look at indignation:
anger or annoyance provoked by what is perceived as unfair treatment.
So I think we can put it together like this.
a persistent feeling of anger or annoyance at a past insult or injury that is perceived as unfair.
And I particularly like this definition because it lines up with my philosophy on why people hold grudges, and what it does to them.

What and Why?

What do grudges do to people, and why do they hold them?

I feel like a large portion of grudges are held because people like being right, and fear being wrong. (Now that I think about it, this is probably why public speaking is one of people’s greatest fears.)

Letting go of a grudge means forgiving someone, which often requires putting yourself in the shoes of others. It does not necessarily mean you were wrong with your actions, but wrong about the other person’s intentions. And when the only feeling of control you may have over an interaction is holding that grudge, it becomes hard to admit “defeat,” or giving up the feeling of control you have. Consequently it becomes very relieving to let go of a grudge and give another chance, because it frees you from a burden (and grudges are large burdens).

What do grudges do to you? Time is eaten up by unpleasant thoughts about someone who you probably have no influence on. (Part of the staying power of grudges is that they are often hard to act upon.) Your relationship with said person is bound to get worse. (If you don’t have a relationship with said person and are mad at them for cutting in line/poor driving/etc, the grudge is even more petty.) Holding a grudge does nothing good for you, and plenty of bad.

The reason for grudges and their effects have already been well-documented; my two cents do not offer anything really special.


Forgiving someone or a group of people whose actions justify a grudge is often one of the hardest things to do. But as numerous examples make clear, letting go of a grudge can be very rewarding. I don’t mean merely mentally rewarding; it often leads to good consequences, or the avoidance of potential disasters. For obvious reasons, any examples I mention will be strictly historical or political (i.e. not personal), so apologies in advance if commentary is a lacks “authenticity”/emotion/etc.

This may be a sore subject even now, but I think the aftermath of the American Civil War is a great example of leniency coming from the victors leading to a positive outcome. For those not in the know, the Confederates could’ve (and by legal standards, should’ve) been tried for treason, and its leaders executed. But that’s not what happened; instead of focusing on inefficiently prosecuting Southern soldiers, the Northern strategy was to use leniency in order to focus on “nation-building.” Historians believe that this may have prevented the South from trying to break again or from majorly impeding the reunification process.

An example of when a lack of leniency and harsh punishments from the “winners” causing disaster is World War I. As a result of unfair punishments, the German population was able to be manipulated into accepting the atrocities of World War II.

In general, famous political bargains are also good examples of when reconciling a grudge leads to better outcomes for everyone involved.

Holding onto a grudge does nobody good; letting go of one has the potential to do good, or at the very least, prevent harm.

Preorders are out!

See here.

The hardcover will be out soon! For now, you can buy it at $7.99 on your Kindle. What a steal! If you haven’t ordered this already, I highly recommend ordering it. I’d order it, except… I already have a PDF of the book for obvious reasons…

Edit: Hardcover comes on April 5th along with the Kindle; there are no preorders.

\input in LaTeX

What not to do

Some people who are not as used to LaTeX write their books the following way:

  1. They write some articles with preambles. (This is fine.)
  2. To compile the whole document, they copy paste their source code and use \part{Chapter 1} \pagebreak to separate each chapter. (And this is if they’re efficient.)

The articles end up being fine, but the whole document is just a jumbled mess.

Here’s the better way.


Here’s what you should do instead.

  1. Make a master document (which we will refer to as master.tex).
  2. Write your articles, but do not include the preamble (that is, do not define theorems, environments, and do not include \begin{document} or \maketitle, among other things). These will be part1.tex, part2.tex… you get the idea.
  3. Do \input{part1.tex}, \input{part2.tex}, and so on.

Then this is how your master document should look:




\part{Title 1}



\part{Title 2}



[… and so on.]


An Example

Let’s give an example of this.

For reference, anything in light blue is optional, though it is recommended for better formatting.





[Use whichever packages you feel like.]



\date{A Fruit}



\part{Title 1}



\part{Title 2}




Notice that there is no preamble, and \begin{document} and \end{document} are missing. This is so \input works.

Did you know that $2+2=5?$


Big Brother is coming for you!

What if I want my articles compiled individually as well?

This is quite easy; keep your articles part1.tex, part2.tex with the preambles.

Then, make a folder called “book” and make copies of your articles. (We will call them copy1.tex, copy2.tex, and so on.)

Inside book, put copy1.tex, copy2.tex, and so on.

Then, master.tex should look like this.




[Use whichever packages you feel like.]



\date{A Fruit}



\part{Title 1}



\part{Title 2}




None of the names chosen (such as “part1.tex,” “master.tex,” and “book”) matter. They can be replaced as desired. I just chose consistent names because 1) encouraging good naming practices is good and 2) for convenience in referring to certain LaTeX files.

(Also, an Anonymous commenter mentions that \usepackage{asymptote} is inferior to \usepackage[inline]{asymptote}. They are completely right about that; I just put some random packages there as “filler.”)

Book – Proofs in Competition Math

This is a book which is about proofs in competition math. This blog post is a promotional one for its release.

Authors: Alex Toller, Freya Edholm, Dennis Chen.

Important Info

Preorders start on March 14th, also known as Pi Day. Yes, this was on purpose. (3/14)

Release is on April 5th. The best way to remember this is you will have to run for your life if you don’t get this book on four-five(4/5)

Why you should pre-order (when the time comes)

First, if you want to see my writing… this book doesn’t have much of it at the time of this post. (Oops.) Unfortunately, I’ve been bogged by other obligations, so I haven’t written much. (If you think the issue of splitting the earnings and me potentially getting money for nothing is an issue, I do too. More on that later.)

But if you agree with what I say on this blog, here’s something you’ll definitely agree with when the book comes out (and even if you don’t agree with me most of the time, you will agree here) – this book is high-quality. I will have proof for that (aka samples) soon, if the other two agree to release nontrivial yet non-significant portions. I’m writing Inversion, so if you liked EEG’s Inversion, you are going to like this one even more.

Also, Alex and Freya are very reputable within the math community themselves (though  I suspect most readers of my blog already know this). (In fact, they’re much more reputable than me, but shh!)

Anyway, this is quality, this is hype, and this covers a lot of stuff.

Actual content

Why you should pre-order: Part 2.

Here’s a rough ToC, with wording that is totally inaccurate.

Part 1: Proofs (this is logic in general, also stuff like iff. Good for beginners.)

Part 2: Algebra

A: algebra for noobs

B: Basic Inequalities + Complex Numbers + just stuff you should know for polynomials

C: Really hard stuff (see newton sums)

Part 3: Geometry

A: normal stuff

B: normal, but more advanced stuff

C: really hard normal stuff + bary/polar/cylindrical + inversion (I’ll get spiral similarity and homothety added if I can)

Part 4: Number Theory/Combinatorics

A: intro+interesting but unimportant stuff (its kind of the equivalent to spiral similarity or inversion)

B: More interesting but unimportant stuff.

C: Classic NT (Bases, mod arith)

D: Hard NT (HELP)

E: Even harder NT. (HELP II)

Part 5: Open Problems + for fun

A: Goes over open problems. Makes no progress but defines the backgrounds. (If we could make significant progress, we’d submit that as an article hm?)

B: Pythagorean’s Proofs (my favorite) and Fake-Proofs (my LEAST favorite).

I’m not being paid for this

I forget if I’ve said this to Alex, but I don’t intend to be paid for this simply due to the fact I’ve done an embarrassingly small amount of work on the book. I will still

a) take responsibility for the final product

b) be involved in marketing

c) actually work on the book now

but the little work I did does not warrant payment. (It’s a wonder I’m on the authors list…) I intend to make up for that by working on it now.

In Conclusion…

Buy the book when it comes out!

I’m a lazy bum too, so I deserve all the blame for flaws because I’d be able to fix them if I was paying more attention and none of the credit for success because I didn’t do anything. (Oops.)

Also, I don’t know where Homothety/Spiral Similarity will go. Preferably next to Inversion, with Homothety before Spiral (since Homothety is a special case of Spiral).

Please support this by sharing with your friends or whoever might be interested!

(If this post seems lazy, it’s because I’d rather get onto writing the book.)

Edit: I’m going to put this in every category, so people see this. I also will update my website soon.


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 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).