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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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/

Quick Update on IGP 2

Hello world,

I haven’t been posting in a while.

We may have another IGP soon. However, we’re also planning to make an entrance exam for next year’s kids (more details on this when we’ve actually started planning some things), so any new problems we create will probably go on the test. However, I usually create some new problems for every IGP, so I’ll probably be using my new problems for IGP and for the Entrance Exam. This causes an issue because I can’t post the IGPs up on geometryexplorer.xyz because I don’t want the exam questions to be findable, even if they aren’t explicitly marked. (For my kids, they will be on the IGP Handouts, if I decide to do it next Tuesday.)

Because of this, the IGPs from now may not be public or only a selection of the IGPs will be public from now on. I apologize for this, but I think the Entrance Exam will be worth the wait.

(P.S. To my kids: Please start working on writing problems! Most of you are very far behind, unless you’ve been working behind my backs, which I hope is the case right now…)

IGP

Now that I’ve actually held an IGP class, I will offer a couple of comments on IGP.

  1. If you’re trying to surprise your students, make their handouts have the same amount of questions. (Have them work alone.) Kids tend to talk a lot, even if you tell them not to. Someone will say “There’s only 5 problems!” and someone else will say “Wait, isn’t there 6?” I tried to lie through this, but it probably wasn’t very convincing…
  2. I’m not a student, so I don’t get the sense of wonder that the students might get from seeing who has the same problems, the whole handout of problems, and can I please just have my problems explained teacher? This will probably apply for someone else too. Nevertheless it was really fun watching them discuss.
  3. I think I made the handouts too hard. Students majorly sucked (compared to how I expected them to do) on the handouts even though I gave formulas. I didn’t expect them to telescope so well though, which is a plus for me…

I feel it is important to reiterate that IGP is not a teaching method, it is a review method!

Handouts for IGP (including individual ones) can be found on geometryexplorer.xyz/igp.

Triangle Centers

We introduced the “four common lines” (altitudes, medians, angle bisectors, perpendicular bisectors) and the “four centers.”

For any of my students who forgot,

Angle bisectors -> Incenter (I)
Medians -> Centroid (G)
Altitudes -> Orthocenter (H)
Perpendicular Bisectors -> Circumcenter (O)

Proof main ideas:

Incenter – Let two of them intersect at I, drop perpendiculars, congruence chase.
Medians – Trivial by Ceva’s (the proof isn’t that informative)
Perpendicular Bisectors -> Let two of them intersect at O, use the fact that AO=BO and AO=CO to get AO=BO=CO. (This is assuming the B/C perp bisectors are the ones that we let intersect at O; it doesn’t really matter.)
Altitudes – Medial triangle + perp bisectors (let them be the perp bisectors of another triangle)

Corollary: Incenter is equidistant from sides.
Corollary: Circumcenter is equidistant from vertices.
Corollary: The 6 centroid triangles (the small triangles made by drawing the three medians) have the same area.

We also went over the proof of O/H isogonal conjugates. This was a simple right triangle angle chase.

The handout/problems are here and here, respectively.

Area of a Garage

In Garage Groupies, we did Area of a Triangle. The handout is available on my website!

Observations: The order I put my lessons in is pretty out-of-whack. I think Circles should come before Roots of Unity (which I apparently didn’t before!) so I have updated the orders to reflect that. I also caught a typo in Problem 1 of Circles; this has been fixed.

I really think 7,8,9 are nice and 10,11,12 are especially good challenge problems. The link to the handout is available here.

Oh, and my TA insists that he is a student.

Thoughts on Barycentric Coordinates

Barycentric coordinates should be learned after Mass Points. This isn’t because Mass Points is an extension of Barycentric coordinates; in my understanding, it is the least frequently used definition of it. So why do mass points come first?

Mass points develop the intuition for barycentric coordinates; instead of thinking of points in relation to an origin, we think of points in relation to a reference triangle. This is true for both mass points and barycentric, and this is quite the hurdle to jump over.

The design of my book reflects this; I consider barycentric and mass points a form of “coordinate bash” the same way as complex numbers, vectors, and Cartesian coordinates are. (I also put trilinear in this category, for those of you who want to learn about it.)

And I’m generous enough to offer it for free, look how nice I am.