As we said earlier, a proof consists of a series of arguments, starting from an original assumption, and designed to show that a given assertion is true. Euclid assumed a set of axioms and postulates. Then, he systematically showed the truth of a large number of other results based on these axioms and postulates. In our study of geometry, we will learn to do the same. We will learn how to construct a proof using only these axioms and postulates, and using results which we have already proved earlier.
Think of Euclid’s Geometry as a huge building with many floors. The foundation stones of this building are the axioms and postulates we have already seen. The truth of these is taken for granted; we cannot prove them from more basic truths. In the building analogy, the foundation level is the lowermost part of the building; there is nothing below it, and everything is built on top of it. The floors above the foundation all exist only because the foundation exists. Similarly, in Euclid’s Geometry, the truth of the various results and theorems we will encounter is based on the truth of the “foundation stones” – the axioms and postulates.
Let us go one floor up from the foundation level of Euclid’s Geometry and see a simple example of the proof of a theorem (a result).
Theorem: Two distinct lines cannot have more than one point in common.
Proof: Suppose that two lines have two points in common, say A and B. However, Euclid’s first postulate tells us that through A and B, exactly one (unique) line will pass. Thus, two distinct lines cannot pass through A and B. In other words, two distinct lines cannot have more than one point in common.
Reflect on this proof for a moment. Notice how we proved this theorem using Euclid’s first postulate. Observe that this theorem is “true” because Euclid’s first postulate is “true”.
Example 1: Prove that an equilateral triangle can be constructed on any line segment.
Solution: An equilateral triangle is a triangle in which all the three sides are equal. Suppose that you have a segment AB:
You want to construct an equilateral triangle on AB. Euclid’s third postulate says that a circle can be constructed with any center and any radius. Now, construct a circle (a circular arc will do) with center A and radius AB. Similarly, construct a circular arc with center B and radius AB. Suppose that the two circles (or circular arcs) intersect at C. Join A to C and B to C:
Clearly, \(AB = AC\) (radii of the same circle) and \(AB = BC\) (radii of the same circle). Also, one of Euclid’s axioms says that things which are equal to the same thing are equal to one another. Thus,
\[AB = AC = BC\]
Thus, we have proved that an equilateral triangle can be constructed on any segment, and we have shown how to carry out that construction.