Give an example of a theorem in Euclidean Geometry that can be proven using synthetic, analytic, and vector techniques. Discuss the implications of instruction using the three techniques for classroom instruction.
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Until the 20th Century, Euclidean geometry was usually understood to be the study of points, lines, angles, planes, and solids based on the 5 propositions and 5 common notions in Euclid's Elements. In the Elements there is no concept of distance as a real number in the sense we know it today. There is only the concept of congruence of line segments (thus one can say that two segments are equal) and of proportion (so that we can say that two segments are in certain proportion to each other); but we cannot say that they have the same (numerical) length. Geometry as studied in this way is usually called synthetic Euclidean geometry and is the subject of Chapter 1 of Geometry: Euclid and Beyond.
Interest in the synthetic geometry of triangles and circles flourished during the late 19th century and early 20th century. One of the best known results of 19th century synthetic geometry is the existence of the nine point circle:
Given any triangle in the Euclidean plane, the midpoints of its three sides, the midpoints of the lines joining the orthocenter (the point of intersection of the three altitudes) to its three vertices, and the feet of its three altitudes all lie on the same circle.
This nine point circle and similar synthetic Euclidean results are discussed in the last section of Chapter 1.
Though numbers as measure for lengths and areas are not explicit in Euclid?s geometry, they are implicit in his arithmetic of line segments (discussed in Chapter 4) and his ?scissors and paste? theory of areas (discussed in Chapter 5). Euclid?s arithmetic of line segments, after defining a unit length determines an ordered field, whose positive elements are the congruence equivalence classes of line segments. Then to develop analytic (or Cartesian) geometry starting from the synthetic Euclidean Plane we choose:
1. A point that we call the origin, O,
2. A segment that we call the unit (length), and
3. Two lines through O, which we call the coordinate axes, (today these are almost always perpendicular, but Descartes did not require them to be so).
We can compare a line segment, a, with the unit and define the length of a to be equal to the ratio of a to the unit. Then both coordinate axes can be labeled as number lines with O being the zero. In the usual way we develop the real Cartesian plane and can now study geometric properties using algebra.
Geometry, as developed in Euclid, was a systematic body of mathematical knowledge, built by deductive reasoning upon a foundation of three main pillars: (1) definitions of such things as points, lines, planes, angles, circles, and triangles; (2) the assumption of certain geometrical postulates regarded as true but perhaps not self-evident; and (3) the assumption of certain axioms or common notions which were taken to be self-evident truths. The body of Euclid's great work consists of a set of propositions or theorems, each derived systematically and logically from the definitions, axioms, and postulates of his foundation and from theorems already proved.
Five postulates may be paraphrased as:
A unique straight line can be drawn from any point to any other point.
A finite straight line can be extended continuously in either direction in a straight line.
A circle can be described with any given center and radius.
All right angles are equal.
If a straight line falling on two straight lines makes interior angles on the same side with a sum less than two light angles, the two straight lines, if produced indefinitely, meet on that side on which the angle sum is less than two right angles.
This fifth postulate is known as the parallel postulate and is essentially equivalent to the statement that "a unique line parallel to a given line can be constructed through any point not on the line."
The develoipment of analytic geometry and utilization of coordinates establishes the
general methods of setting and studying the lines. The analytic expression of a curve is its
equation, and the visual representation is the graph of the function represented by the
equation. The presence of the coordinate system, with which the curve is not naturally
related is a drawback to analytic research. The coordinate system with its properties, ...