By 苏剑林 | March 17, 2015
Euclid
The theme of this article is parallel lines. Friends who are familiar with mathematics might expect me to write about non-Euclidean geometry. But not this time. The content of this article is purely about the Euclidean geometry we have studied since childhood, based on "Euclid's Fifth Postulate" (also known as the Parallel Postulate). However, even within the Euclidean geometry we've studied for so long, there are many problems regarding parallel lines that we might not have thought through clearly. Because parallelism is such a fundamental concept in geometry, discussions of such basic propositions are quite prone to circular reasoning or even putting the cart before the horse.
Starting from middle school, we are indoctrinated with rules for judging parallel lines such as "if corresponding angles are equal, the two lines are parallel" and "if alternate interior angles are equal, the two lines are parallel." Of course, we cannot forget: "through a point outside a line, there is exactly one line parallel to the given line." However, among these concepts, how many are fundamental axioms and how many can be proven? How should they be proven? I think many people do not have a clear understanding of this, and I myself did not have a very good answer. Even the teachers who teach parallel lines in middle school likely cannot explain it clearly. Later, I discovered that I actually did not know how to prove "if corresponding angles are equal, the two lines are parallel." "Euclid's Fifth Postulate" doesn't seem to tell us this judgment rule directly. So, I looked back at my middle school mathematics textbooks and found that the rule "if corresponding angles are equal, the two lines are parallel" was given to us to accept without proof. No wonder I could never think of a simple proof for it...
Therefore, I wanted to write this article to provide some reference for understanding the entire logic of parallel lines.
Finding the Parallel Line
At the outset, we declare that we are only discussing Euclidean geometry. Therefore, we accept the following axiom:
Through a point outside a line, there is exactly one line parallel to the given line.
Of course, in Euclid's Elements, there are four more fundamental axioms:
1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are equal to one another.
In fact, the first axiom implies that through two points only one straight line can be made, and the second axiom implies that all straight lines have infinite length. Without these "implications," the above axioms would not be sufficient to constitute the foundation of Euclidean geometry. Additionally, it is not difficult to prove that "through two points only one line can be made" is equivalent to saying "two different lines in a plane can have at most one intersection point."
The above content is the foundation of this article and, of course, the foundation of Euclidean geometry. First, the Fifth Postulate says there is only one parallel line. So let us first find it. The method of finding it is as follows:
The green line is the given line, the purple point is the given point. Construct a perpendicular line to the given line through the purple point.
Note that for every step we take, we must consider its basis to avoid circular reasoning. Why can we construct a perpendicular to a given line? First, we draw an arbitrary line through the purple point to intersect the green line, resulting in two angles $\angle A$ and $\angle B$, where $\angle A - \angle B > 0$. When the line rotates around the purple point to the right, there must eventually be a position where $\angle A - \angle B < 0$. Thus, there must be a position where $\angle A = \angle B$. Since $\angle A + \angle B = 180^\circ$, then $\angle A = \angle B = 90^\circ$. This is essentially the Intermediate Value Theorem for continuous functions—content from mathematical analysis! You read that right; even such a simple problem requires it. If geometry is to be strictly formalized, it must rely on algebra as a tool!
Next, through the purple point, we draw a line (the blue line below) perpendicular to the orange line constructed in the previous step.
Now we can prove that the blue line is parallel to the green line. The idea of the proof is very simple: the entire figure is symmetric with respect to the orange line. If the blue line and the green line intersect on one side, they must also intersect on the other side, resulting in two intersection points—which contradicts the axiom "two different lines in a plane can have at most one intersection point." Therefore, the two lines must be parallel.
We have now found one parallel line, and according to the Fifth Postulate, there is only one such line, so this line is the unique one.
"If Alternate Interior Angles are Equal, the Two Lines are Parallel"
Next, let's prove "if alternate interior angles are equal, the two lines are parallel." In my opinion, this is not simple at all...
First, we have two parallel lines: the blue line and the green line. The yellow line is the transversal intersecting the parallel lines, resulting in two alternate interior angles $C$ and $D$. Next, through the intersection point at angle $C$, draw a pink line perpendicular to the blue line. It can then be proven that the pink line is also perpendicular to the green line. This step seems obvious, but it also requires proof. If the pink line were not perpendicular to the green line, then by using the previous method, we could construct another line passing through the same point that is parallel to the blue line, which would mean two different lines through the same point are parallel to the blue line, a contradiction. Furthermore, we can easily prove that the red line is parallel to the pink line.
Now we have obtained a rectangle—a figure with four interior angles of $90^\circ$. Thus, we can use the symmetry of the rectangle to prove that two triangles are congruent (i.e., triangles with three corresponding sides equal are congruent), thereby proving that the alternate interior angles are equal. This judgment rule originates from the stability of triangles—which is also a geometric axiom.
But don't forget, we haven't yet proven that the opposite sides of a rectangle are equal!! Here, a rectangle is defined as a quadrilateral with four $90^\circ$ interior angles; thus, it does not inherently include the condition that opposite sides are equal, which we need to prove. Of course, it is not difficult. Using symmetry, fold the rectangle over; according to our construction of parallel lines (two perpendiculars), the parallel line at the midpoint is the axis of symmetry, so they overlap after folding. (This is stated colloquially, but it can be written in mathematical language—readers, please try it yourselves; symmetry is the key.)
A Brief Summary
To explain the matter of parallelism, we have spent quite a bit of space, and I am still not sure if I have made it entirely clear. Therefore, when fundamental issues like these are involved, one must proceed step by step with questioning to avoid falling into logical contradictions. Of course, whether it is necessary to go to this length is a matter of opinion.
There may be places where my reasoning is not entirely clear or where logical contradictions might occur. I hope readers will not hesitate to offer criticism should they find any.