Lesson's in Geometry
Geometry is a subject in Mathematics that has been around for ages. One of the earliest people to write down the principles of Geometry was the Greek philosopher Euclid. In the mid 1800's, John Playfair translated Euclid's works into English .
The works of Euclid begin with some basic definitions. Using the Geometer's Sketch pad, I have illustrated these definitions.
In addition to the definitions, Euclid spelled out what Playfair called postulates and axioms. These provide a framework for what has been come to be known as Euclidean geometry.
The works of Euclid are separated in several different "books." Within each book are propositions about geometrical objects. A proposition can be a statement of some truth that can be proven. This is called a theorem. A proposition can also be a problem. This is a way to construct or draw a geometric figure. Both can be proven to be true. The proof uses the definitions, postulates, axioms, and propositions already proven. The following lessons are based on these two types of propositions.
- For theorems, we use a figure we have drawn and go through the step by step proof of what the theorem says.
- For problems we use GSP to give the step by step procedures for constructing a geometric object. Then we go through the step by step proof of these constructions.
Proof of construction for an equilateral triangle
Proof of construction of equal straight lines
Proof of construction of part of a line equal to a shorter line
Lesson 11 Propositions IX through XII are construction problems. The reader is encouraged to construct the proof of these on his or her own.
There are many more constructions and theorems in geometry. I hope you have profitted from the ones here and will explore others.
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