Euclidean Geometry
(Essential Elements)

Right Triangle Similarity
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Parallelogram from Triangle Duplication

 Deductive logic in maths may begin
  here.  But deductive   logic mastery 
itself  may begin with words and stories.

Complex Numbers
Update (December 13th, 2009). A
simpler & quicker development of 
complex numbers  is  available: 

Similarity of Right Triangles

Two right triangles are similar if they have an angle in common BESIDES a right angle. More generally, two triangles are similar if and only the angles in one equal the angles of the other in some order. But that topic is not of interest today. 

For similar triangles, the ratio of their matching or corresponding sides are equal.  The diagram below illustrates this in a special case.

The observation or assumption that

the ratios of side lengths, here adjacent/hypotenuse or opposite/hypotenuse are independent of the scale factor and depend only the angle the hypotenuse makes with the adjacent side

justifies the definition and calculation of trig functions using the ratio of sides for similar right triangles.  See below.

Trigonometric Functions of Acute Angles

Given a right triangle with acute angle , say 

we may form the ratios

The following chains of reason show that these ratios depend only on the angle and are independent of the scale or size of the triangle we use to compute the ratios.

Why computation does not depend on scale.

Suppose we have two right triangles

with a common angle as shown.  

By the AA minimal condition for similarity of triangles, the presence of the common angle  in addition to the common right angles in each triangle implies the two triangles are similar. Therefore, there is a proportionality constant K such that

a = Kd, b= Ke and c = Kf.

Therefore the ratio of adjacent over hypotenuse 

coincides in each triangle. So this ratio does not depend on the right triangle from which it comes provided there is a common angle .

Like wise the ratios of opposite over hypotenuse

and opposite over adjacent

do not depend on the right triangle from which they comes provided there is a common angle .

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Math How-TOs
1. Arithmetic   2. Algebra   3.  More Algebra  4.  Geometry 5 More Geometry 6.  Calculus
>> densely written 
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Online Volumes (orders)
1,  Elements of Reason. 1996
1A. Pattern Based Reason  1995
1B. Math Curriculum Notes 1996
2. Three Skills for Algebra  1995
3 .Why.Slopes.&.More.Math.1995

Skill & Concept 
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 1. Decimal Arith - Video Based ]
2   Fractions  
3.  Fractions  with Units  
3. Solving Linear Equations  - 
making alg easier
4. Formulas forwards & Backwards - unifying theme for Algebra
5.  Proportionality, Back- & For-wards - theme at work.
6.  Logic - Math Free, good for precision in  work & studies 
7. Euclidean-Geometry  (leanly)
8. Slopes and Lines 
9. Why Study Slopes - a context 
10.  Quadratics
11  Polynomials
12  Factored Polys - a context
13 Functions - For-& Back -wards
14  Number Theory, Richly
15. Exponents, Radicals & logs.  
16   Calculus - Examples & Advice 
17.   Real  Analysis 
18  Electric Circuits Etc (So So)
19 Maps, Similarity & Trig, (alt view)
20 Complex numbers  
21 Logic with Symbols+truth tables
22  Consistent Story Telling
23. Even More Logic