While maths is all around us, lying at the heart of the physical world, the underlying ideas and concepts we are familiar with lie in the abstract world. When it comes to proof in mathematics, we must search for a definitive method of showing that something is true, and we cannot simply turn to observations around us. This is the fundamental difference between science and mathematics – mathematics searches for something more profound, something we can be absolutely sure of. However, sometimes we need to find a way of representing physical objects in a more abstract manner and this is where geometry arises. But sometimes, we must additionally find a way to represent physical manipulations, such as motion, and perhaps a good way to introduce these ideas is by attempting to prove that a triangle is isosceles when its base angles are equal, using motion.

Introduction

Most young mathematicians may be familiar with various geometric proofs that if the base angles of a triangle are equal, then it is isosceles. By intuition

however, it seems obvious that if we could manipulate the triangle such that we place the left leg in place of the right leg and the right leg in place of the left leg, we would have a proof that this triangle is isosceles. We can carry this out using motion – turning the triangle over. This seems obvious in the physical world: we can simply take a triangle-shaped piece of paper and turn it over. However, trying to make sense of this in the abstract world is problematic. Zeno’s paradoxes on motion perhaps best illustrate this. One of them states that in order to move from a point A to a point B, it must move to a point C in between. But to move to that point C, it must move to a point D and so on. So we hit a brick wall – motion seemingly can never begin (in the abstract world).

Making Sense of Motion

However, we need only ask – is there any need to consider the movement? Perhaps it doesn’t matter where the triangle goes in between the movement so long as we know where the triangle ends up at the end of the motion. For us to have this knowledge, we must know where each individual point of the triangle ends up and so we must have some way of making a record of these points. A very simple and familiar way to do this is to employ the Euclidean plane. We can place a triangle on the plane and specify a motion (say 2cm right and 3cm up). If one of the points of the triangle was (0, 0), it’s new position would be (2, 3). Once we apply this to all the points, we then have two triangles on the plane, and there has been no movement. The original point (0, 0) has not moved. What we have instead done is move our attention to a new set of points. We define this change of attention with the coordinates of the new triangle.

Choosing Which Motion to Make

Above was an example of rigid motion. Rigid motions are movements that do not change the shape or size of an object. We need such a motion to represent the flipping of the triangle as we are trying to prove that the original triangle and the triangle after the motion are the same, meaning we must keep the shape and size the same. We can easily see that the particular motion we need is a reflection, specifically a reflection along the altitude going through the apex (what would be the perpendicular bisector of the apex).

Completing the Proof

Now we can complete our proof. We define a triangle with equal base angles on the Euclidean plane. We then reflect it along an altitude which we say passes through the apex of the triangle. The equality of the base angles implies that, without loss of generality, by reflecting the triangle along this line, the reflected triangle should lie exactly on top of the unreflected triangle. Therefore, we prove that the triangle is isosceles.

Finishing Note

Motion is fundamentally a physical idea and to simply take experimental evidence does not necessarily prove something, it merely constitutes a theory. We cannot simply make a cardboard cut-out of a triangle with equal base angles, turn it over, notice that it ends up exactly where it was before and call this a proof. We have no way of generalising this for all triangles with equal base angles, so this remains a theory, not a proof. Using coordinate geometry, we can apply this idea using abstract methods and, in the process, by using mathematical reasoning we can indeed begin to construct proofs. Finally, we can apply this idea even further, such as using rotation to prove equal arcs of a circle give equal chords.

Reference

I. Stewart. Concepts of Modern Mathematics. Dover Publications. 1995.