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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.
- Correspondence between triangles.
Here is an explicit definition, not always seen in class.
- Isometry of Triangles - Here is a
definition.
- Side-Side-Side (SSs) method for
triangle construction and SSS like method for locating point.
- Side Angle Side (SAS) method
plus an application
Ruler and Compass Construction to Bisect an Angle
- Angle-Side-Angle (ASA)method,
and ASA-like method for determining current location in navigation.
- 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)
- Side-Side-Side Failure
- SAS Failure or Near Failure
- ASA Failure - links
with the parallel postulate
- Parallel Lines - and
angles associated with a transversal.
- Triangle Angle Sum - from the
parallel postulate
- Similarity and Minimal
Conditions for
- Right Angle Trig.,
from Similarity
- Trig & Similarity
- More about the Connection
- Parallelograms and their Properties
- Kite Construction
from triangles
- Parallelogram Construction
from triangles
Links:
- 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 ..
- TopStudy More Geometry
More Seven More Interactive (step by steps) online proofs
- 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.
- 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).
- 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.
- Distributive
Law, Step I -
- scaling distributes over addition. The proof here may employs similarity concepts
- Distributive Law,
Step II -- rotation distributes over addition. How rotation via angle distributes over addition. The
proof here uses triangular isometry arguments.
- 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.
- Addition of points in the plane
(Vector Addition Preliminary)
- Multiplication of Points in the Plane
with notation for complex numbers and basic rectangular coordinate,
product formulas.
- Distributive Law, Step I
- scaling distributes over addition.
- Distributive Law, Step II
- rotation distributes over addition
- 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
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.
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.
Properties of parallelograms follow and combine earlier properties of
triangle construction or isometry criteria and the properties of parallel
lines.
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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Euclidean Geometry
with a geometry based
based development of
complex numbers
24 Lessons:
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
Easy Consequences of this (newest) Complex
Number. Starter Lesson in this site folder follow below.
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
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