exploring euclidean geometry 2: electric boogaloo

First, as a teaser, I’ll publish a couple of things on here.


It is very easy to teach geometry wrong. There are plenty of high school geometry
books written for profit that do exactly this. Rarely are underlying connections pointed
out, or are thought-provoking problems required. More often than not, those who do
not enjoy math associate it with brutal calculations with ugly numbers. Rounding to the
nearest hundredth or to the nearest inch, and messing up the calculations has been a
constant source of frustration for those in typical math classes.

But the art of math is filled with beauty. In reality, most ratios are simple. A surprising
fact is that in a triangle, HG = 2GO. Most configurations will have beautiful numbers
involved, and each problem’s numbers, if any, have been chosen with care. Rarely will a
problem require unreasonable computation, because understanding the concept will
be enough to solve it.

While these theorems may seem memorization based at first, a deeper understanding
of each theorem and enough practice with them on significant, thought-provoking
problems will be more than enough to have them ingrained in your head. Putting
formulas on flashcards, writing down a formula hundreds of times, and so on are not
efficient methods because the best way to learn to solve problems is to do them!
However, if a problem does not provoke thought, it will not stick around in your brain.
Only when our brains are sufficiently stimulated will they be able to remember what
stimulated them in the first place.

Geometry can be taught wrong in so many ways, but there are so many ways to teach it
right. Only through a variety of perspectives will you understand a concept deeper.
This is one of the many resources out there, and other good resources should be used
as well.

With that said, let us begin Exploring Euclidean Geometry!

Table of Contents

A. The Fundamentals
1. The Axioms of Geometry
2. Definitions and Properties
3. Logic in Mathematics

B. Playing with Circles
1. Circles and Angles
2. Circles and Lines
3. Radical Axes

C. Triangles
1. Area of a Triangle
2. Concurrency and Collinearity
3. Lengths in a Triangle
4. Circles and Triangles

D. Trigonometry
1. Sine, Cosine, and Tangent
2. Reciprocals and Inverses
3. Trigonometric Identities
4. Graphing Trigonometric Functions

E. Analytic Geometry
1. Cartesian Coordinates
2. Conic Sections
3. The Complex Plane
4. Vectors and Matrices
5. Mass Points
6. Barycentric Coordinates
7. Trilinear Coordinates
8. Miscellaneous Algebraic Problems

F. Transformations
1. Generic Transformations
2. Homothety
3. Similitude
4. Inversion
5. Isogonal and Isotomic Conjugates
6. Projective Geometry

G. The Third Dimension
1. Volume and Surface Area
2. Lengths and Cross Sections

H. Extra
1. Pythagorean Theorem
2. Constructions
3. Directed Angles
4. Geometric Inequalities
5. Additional Problems

Progress Report?

What’s in green has already been written, and what’s in red has yet to be written. Fortunately, I only have 2 more beasts to tackle: Projective Geometry and Geometric Inequalities. Pencils/harmonic bundles should not be very hard to cover. Geometric inequalities stem from geometric equalities + inequalities, so it shouldn’t be too hard. Additional problems are for Alg/Combo/NT that I like.

So when will this be done? The short answer is I don’t know. I’m going to be busy for the fall of Freshman year ( marching band 😦 ) and it takes motivation to write the book instead of doing other things. However, I’m optimistic it could be finished in 2 months – but I don’t think I want to rush it or full effort for 2 months. The current pagecount is 306, and I’m expecting to hit at least 375. 400 might be realistic.

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