Euclidean Geometry plus a Geometric Development of Complex Numbers

The development of Euclidean Geometry here is very simple. It only employs implication rules A if B directly, one at a time and one after another to provide explanations easily understood and repeated.  

The Complex Number and vectors in the plane supplement applies the results and assumptions of Euclidean geometry to imply or suggest their arithmetic properties.  

Euclidean Geometry Leanly
(Geometry without 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 may be  employed in the development of geometry with coordinates.

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. Common Terms and Vocubulary:   Points, lines, rays, line segments. 
2. Correspondence between triangles. Here is an explicit definition, not always seen in class. 
3. Isometry of Triangles - Here is a definition.
4. Side-Side-Side (SSs) method for triangle construction and SSS like method for locating point.
5. Side Angle Side (SAS) method plus an application
   Ruler and Compass Construction to Bisect an Angle
6. Angle-Side-Angle (ASA)method, and ASA-like method for determining current location in navigation.
7. Isosceles  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)
8. Side-Side-Side Failure 
9. SAS Failure or Near Failure 
10. ASA Failure - links with the parallel postulate
11. Parallel Lines - and angles associated with a transversal.
12. Triangle Angle Sum - from the parallel postulate
13. Angles in Circles  Easy Consequences of Properties of Isosceles Triangle  
14. Drawing Circles through the vertices of Triangles with right bisectors
15. Similarity and Minimal Conditions for
16. Trig Ratios for Similar Right Triangles
17. Trig Functions from Trig Ratios - More about the Connection
18. Parallelograms and their Properties
19. Kite Construction from triangles
20. Parallelogram Construction from triangles

See annotated guide below for the above steps in geometry.

Annotated Guide

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.   

Construction and Congruence

The issues of triangle duplication and Isometry  via  the triangle construction methods and isometry criteria (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. 

• When a method works, the resulting triangle is isometric to any triangle drawn with the same method and data.
• Isosceles  and Equilateral Triangles  may be described in different (equivalent) ways. That follows from isometry criteria (SSS, SAS and/or ASA)

Each triangle construction method 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.  

Parallel Lines 

Describing when ASA Fails points to and provides a context for the parallel line postulate - and correspond Euclid's form of it.   The latter represents here an extrapolation of experience with the ASA triangle construction method. 

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
• Angles in Circles - Triangle angle sum and recognition of Isosceles triangles lead to results. 

Similarity and Trigonometry.

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 

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

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