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.

Projective

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).

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