MATH 3581College

Geometry

D. P. Dwiggins, PhDDepartment of

Mathematical Sciences

Course Outline

• Part I = Euclidean Geometry• Circles and Triangles• Concurrence, Collinearity, and Concyclicity• Constructions

• Part II = Projective Geometry

• Part III = Noneuclidean Geometry• History of the Parallel Postulate• Spherical Geometry and Trigonometry• Hyperbolic and Elliptic Geometry

Synthetic Geometry

Analysis: the act of taking something apart to see how it works

Analytic Geometry: use a coordinate system to assign numbersto every part of a geometric object, determine which algebraic equations are used to relate these numbers, and use the equationsto deduce properties of the geometric object

Synthesis: the act of making the parts fit together to form the whole

Synthetic Geometry: begin with a stated list of axioms and postulates concerning geometric space, and use rules of logic and the formation of syllogistic arguments to deduce properties of the objects in geometric space

Analytic Geometry vs

Beginning in the 1960’s, since analytic geometry is used to illustrate the development of concepts in calculus, colleges began referring to their calculus courses as “Calculus and Analytic Geometry.” However, the only real geometry being done involved equations for lines, circles, and conic sections.

The intent was for high school students to study synthetic geometry, basically by following Euclid’s text. Since college-bound students were also supposed to be reading Homer and classical texts in Latin, it was anticipated that first-year college students would be well-based in classical literature, philosophy, and mathematical development.This of course never came anywhere close to happening.

For both of these reasons, a new course in College Geometry was created at the U of M in the early 1990’s, and other universities across the nation have begun doing the same.

Synthetic GeometryAnalytic Geometry vs

GeometryHigh School

A typical high school geometry course outline would be to work through the material found in Euclid I:1-48, which begins with the construction of an equilateral triangle and concludes with the Pythagorean Theorem.Along the way, triangle congruence theorems are used to deduce properties of circles and quadrilaterals.Finally, the postulate of measure is introduced and students learn how to calculate areas, volumes, and proportions in geometric figures.Unfortunately, in modern times the emphasis is on learning the material needed to answer questions on the ACT, and most of this material is quickly forgotten once the test is over.

on the ACTGeometry

• Definitions• Lines and Angles• Polygons and Triangles

• Polygons• Similar Triangles• Quadrilaterals• Areas of Irregularly Shaped Regions

• Solids• Surface Area and Volume• Prisms, Pyramids, and Spheres

• Circles• Central and Inscribed Angles• Secant and Tangent Lines• Chords and Arcs

for Parallel LinesAngles on a Transversal

Corresponding Angles are CongruentVertical Angles are Congruent

Alternate Interior Angles are CongruentSame-Side Interior Angles are Supplementary

et cetera

1 2

3 4

5 6

7 8

QuadrilateralsRectangle

Equiangular, perhapsnot equilateral

RhombusEquilateral, perhaps

not equiangular

KiteDiagonals areperpendicular

TrapezoidTwo sides are parallel,but not the other two

Circles and Angles

Central AnglesVertex at center

80º

35º

A

B

P

Q

Inscribed AnglesVertex on circumference

80m AB

70mPQ

Circles and Lines

Euclid III:36 states the circle divides the secant externally in a product giving the same as the square of the tangent.

A

B

C

D

2That is, .AD AC AB

Tangent Line: touches

the circle at one point ( )

, to touch

Secant Line: cuts the

circle at two points ( , )

, to cut

AB

B

tangere

AC

C D

secare

�������������� �

�������������� �

Circles and Lines

Actually, here is how Euclid III:36 is stated, when translated from Greek into English:

A

B

C

D

If a point be taken outside a circle and from it there fall on the circle two straight lines, and if one of them cut the circle and the other touch it, the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference will be equal to the square on the tangent.

In Greek mathematics, the only way they made sense of the product ab of two positive numbers a and b was to interpret it as the area of a rectangle with sides of length a and b. This is why the product aa = a2 is called the square of a number.

Harmonic RatiosA

B

CDE

Here is how Euclid III:35 is stated:

If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.

Remembering how the Greeks viewed products in terms of the areas of rectangles, write down what this statement means in terms of the labeled parts in the diagram. Why is this statement trivial if E is the center of the circle?

Is it possible to use this result to obtain other results?For example, is there any way to obtain a relation between AB and CD?What happens if AB CD? (See Euclid III:3-4.)