Part I. Euclidean Geometry Leanly
(Geometry Before Coordinates) 

Euclid about 300 BC in his elements produced a codification of geometry before the invention of coordinates by Renes Descartes 1800 year later.  Knowledge of Geometry before coordinates is employed in the development of geometry with coordinates (analytic geometry, unit-circle trig, complex numbers, calculus, and so on).

This area on Euclidean Geometry  on geometry before coordinates offers thought-based explanation of the following.  Try to read them in sequence. There is more to Euclidean Geometry than this, but the following elements cover the least amount possible for the following site development of analytic geometry and trigonometry.

1. Correspondence between triangles. Here is an explicit definition, not always seen in class. 
2. Isometry of Triangles - Here is a definition.
3. Side-Side-Side (SSs) method for triangle construction and SSS like method for locating point.
4. Side Angle Side (SAS) method plus an application
   Ruler and Compass Construction to Bisect an Angle
5. Angle-Side-Angle (ASA)method, and ASA-like method for determining current location in navigation.
6. Isoceles  and Equilateral Triangles plus applications: Construction and Characterization of a Right Bisector of a Line Segment  and Ruler and Compass Construction of a Perpendicular from a Point to a line (with properties of such perpendiculars)
7. Side-Side-Side Failure 
8. SAS Failure or Near Failure 
9. ASA Failure - links with the parallel postulate
10. Parallel Lines - and angles associated with a transversal.
11. Triangle Angle Sum - from the parallel postulate
12. Similarity and Minimal Conditions for
13. Right Angle Trig., from Similarity
14. Trig & Similarity - More about the Connection
15. Parallelograms and their Properties
16. Kite Construction from triangles
17. Parallelogram Construction from triangles

Links:

1. Top Study Geometry:  Seven Interactive (step by steps) online proofs of (1) vertically opposite angles are equal, (2) Sum of angles in a triangle = 180 degrees (3) equality of angles at base of an isosceles triangle ..
2. TopStudy More Geometry More Seven More Interactive (step by steps) online proofs
3. Top Study MATH Link  Visit here for Arc, Area and Volume Calculation (Mensuration) formulas

Part II. Complex Numbers Geometrically

All field properties of the complex numbers except for one, a distributive laws are simple consequences of  an extrinsic geometric definition of addition and multiplication of points in the plane using rectangular and polar coordinates, the assumption of the equivalent of the latter in determining points in the plane, and the field properties of real numbers.  Site coverage of arithmetic derives the latter field properties in an extrinsic (geometric)  manner  

This treatment (August 20, 2008)  provides a definition and properties via an application of Euclidean  geometry and the properties of real numbers. The aim is to prove the distributive law for multiplication over addition as all other field properties of the complex numbers are easy consequences of definitions, an assumed equivalent of rectangular and polar coordinates, and field properties of real numbers. Details follow in the next five lessons. 

1. Addition of points in the plane - Say or define how to compute sums and how the origin, the summands, and  the result provide the vertices of a quadrilateral with opposites sides equal in length (at least when the summands and origin are not collinear).
2. Multiplication of Points in the Plane. Say or define how to use the polar coordinates of a pair points in the plane to define a product. Introduce complex numbers and derive a few key formulas for the expression of products in terms of rectangular coordinates. Includes  notation for complex numbers and basic  rectangular coordinate, product formulas.
3. Distributive Law, Step I - - scaling distributes over addition. The proof here may employs similarity concepts
4. Distributive Law, Step II  -- rotation distributes over addition.  How rotation via angle distributes over addition. The proof here uses triangular isometry arguments. 
5. Distributive Law, Step III  rotation & scaling together distribute over addition. That gives a proof of the distributive law for complex numbers. It further implies product formulas in terms of real and imaginary parts.  
   
   This simple introduction of complex numbers relies on the distributive law to complete the derivation and description of the field properties of complex numbers. Ignore references in it (written earlier) to the question of how to derive the distributive law. 

  The site area on Complex Numbers gives easy consequence for calculations involving unit circle trigonometry and vector products (dot and cross). Easy consequences include another proof of the Pythagorean theorem if the derivation of the distributive law, step I, relies on similarity theory for triangles. 

1. Addition of points in the plane (Vector Addition Preliminary)
2. Multiplication of Points in the Plane  with notation for complex numbers and basic  rectangular coordinate, product formulas.
3. Distributive Law, Step I  - scaling distributes over addition.
4. Distributive Law, Step II - rotation distributes over addition
5. Distributive Law, Step III - rotation & scaling together distribute over addition. That gives a proof of the distributive law for complex numbers. It further implies product formulas in terms of real and imaginary parts. 
   
   This simple introduction of complex numbers relies on the distributive law to complete the derivation and description of the field properties of complex numbers. Ignore references in it (written earlier) to the question of how to derive the distributive law. 

To learn more about complex numbers and easy consequences of having two different ways to compute products, the first with polar, the second with rectangular, see the   Complex Numbers  site area.  This August 20, 2008, is preceded by several other developments in site pages of the complex numbers and their properties. The motivation for all developments comes from a simple description of physics as the addition and multiplication of arrows in the plane in several minutes of 1979 public presentation of the late Richard Feynman, a physicist second to none. 

Comments

The the hand-waving and thought-based development of geometry without coordinates in this section  is written by a student of geometry, one who not read Euclid's Elements as is or in translated form, but has only seen shadows in my high school days and other works on geometry.  What remains to be done is to   compare and contrast the treatment here with Euclid Geometry as originally presented in 10 Volumes and various high school shadows there-of.

Correspondence between triangles are often used in the early discussion of isometry and similarity without any definition. So we begin with that.   

The issues of triangle duplication and Isometry  via  the triangle construction methods and isometry critiria (SSS, SAS and ASA) is separated from whether or not the data for the corresponding construction methods work

• Side-Side-Side   
• Side Angle Side   
• Angle-Side-Angle

Lengths and angles must satisfy some inequalities before the methods work.  Those inequalities are automatically satisfied by data coming from an existing triangle. 

Isosceles  and Equilateral Triangles  may be described in different (equivalent) ways. That follows from isometry critiria (SSS, SAS and/or ASA)

Each triangle construction methods may fail. See when  has some consequences.

• In constructing a triangle from three lengths, the  Side-Side-Side Method Fails when and only when the longest length is greater than the sum of the other two. See the discussion of the triangle inequality.
• The SAS Failure or Near Failure occurs when the included angle is two right angles or the included angle is larger than two right angles. The first case gives a flat triangle while in the second case the included angle is external to the triangle and not interior to it.  

In constructing a triangle from angle-side-angle, we observe (or assume) the method will work when and only when the sum of the angles is less than two right angles.  Describing when ASA Fails points to and provides a context for the parallel line postulate.  The latter represents here an extrapolation of experience with the ASA triangle construction method. History Buffs: How close is this view to origins of the or a parallel postulate of Euclid? 

Properties of parallel lines, in particular the angles formed by transversals are developed next. The latter imply the sum of angles in a triangle is 180 degrees or two right angles.

• Parallel Lines - and angles associated with a transversal.
• Triangle Angle Sum - from the parallel postulate

The classical development of right triangle trigonometry then follows from similarity.  We see how trigonometry hides similarity considerations and gives an alternative to them solution of missing side and angles problems for triangles. Similarity is implicit in trigonometric computations.

• Similarity and Minimal Conditions for
• Right Angle Trig., from Similarity
• Trig & Similarity - More about the Connection

Properties of parallelograms follow and combine earlier properties of triangle construction or isometry criteria and the properties of parallel lines. 

• Parallelograms and their Properties
• Parallelogram Construction Methods 

Plane Theory of Vectors and Complex Numbers 

This theory is built on items 1 to 15, and the assumption or practice that rectangular and polar coordinates can be used interchangeably in the plane to locate points permit a definition of point addition using (relative) rectangular coordinates and point multiplication using (relative) polar coordinates.  The coordinates are relative to a choice of unit length. Steps Follow.

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4. Euclidean Geometry

Next

Area Intro
Correspondence
Isometry
Side-Side-Side
Side Angle Side
Angle-Side-Angle
Isoceles
Right Bisector Construction, Etc.
Perpendicular - Point to Line
SSS Failure
SAS Failure
ASA Failure
Parallel Lines
Angle Sum
Similarity
Right Triangle Similarity
Trig  or Similarity
Parallelograms
Kites From Triangles Duplication
Parallelogram from Triangle Duplication
Addition of points in the plane
Multiplication of Points in the Plane
Distributive Law, Step I
Distributive Law, Step II
Distributive Law, Step III

Above Average Students in Geometry may enjoy  the site geometric introduction of complex numbers and the wordy volume 1A,  Pattern Based Reason.

For algebra, logic starter lessons,  see Volume 2, chapters 1 to 12, plus 14, 16 and 17.

Analytic Geometry, Functions & Trig

(FN) What are Functions?
(FN) Functions - More
SZM: Sign, Zero, Monoticity
(L) Lines Summary
(P) Polynomials (*,+,-)
(Q) Quadratics
(D) Simplify Square Roots
(T) Unit Circle Trig
Conic Sections

More For Analytic Geometry:

Real Numbers
Say More Positive
Linear Inequalities
Triangle Inequality
Absolute Value |x|
|x| Eq'ns & Inequalities
Rectangular Coords
Shortest Path
Distance Formulas
Add & Multiply Points
Polar Coordinates
Radians
(A) Vectors
(A) Coordinate Arithmetic
(A) Navigation on Maps
(A) Addition Geometrically
(A) Rotation
(PT) Translations
(PT) Dilatations
PT: Rotations

Easy Consequences of  this (newest) Complex Number. Starter Lesson follow below to provide an alternate development of HS or college maths.

Vec & Cmplx  No Applet
B2 C. Conjugates
B3 Pythagoras
B4 Distance
B5 Rt Triangle Similarity
B6 Trig., Functions
B7 Dot & Cross Products
B8 Cosine Law
B9 Exponential & cis fns
B10 Easy Trig Identities
B11 Set Viewpoint

Lesson Plans and lessons

Secondary I - fractions & allied concepts (decimals, percentages)

Secondary II - Algebra  (arithmetic versus algebraic methods, backward use of formulas and proportionality equations)

Secondary IV - Functions to Trig & Statistics

Calculus Intro 

Algebra Lesson Notes - All levels

Great_Expectations: If you can learn to follow a multi-step methods in any subject precisely, you can do so in other subjects, as well.

Good news: Site pages  identify what you need to study.

Bad news: Site pages do not explain everything  

Worse news: Learning takes time, yours

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