Short Course on Euclidean Geometry
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.
Common Terms and
Vocubulary: Points, lines, rays, line segments.
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.
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.
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.
Properties of parallelograms follow and combine earlier properties of
triangle construction or isometry criteria and the properties of parallel
lines.
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