MATH325: Lecture 2


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Proof of Euclid's Fifth Postulate

Recall that Euclid's goal, and ours, was to use a very few elementary assumptions as the basis for a proof of all of the theorems he already knew to be true. Our first example of a geometric proof will be the proof of Proposition 5, the Pons Asinorum. We'll go through this proof relatively quickly and later see the different method of proving it presented in the textbook.

What we'll prove is that if the two sides of a triangle are the same length, the angles those sides make with the third side must also be equal. Euclid proves slightly more than this, because he can.

Euclid first introduces and names some new points to help in the proof. Points D and E are just points "further down the line" and don't figure in the rest of the proof. Point F can be any point on line AD, and G is a point on line AE chosen so that |AF| = |AG|. (How do we know there is such a point? Postulate 3 allows Euclid to define a circle with radius |AF| and center A. Euclid then assumes (without justification) that that circle will intersect line AE at some point, which he names G.)

He then uses Proposition 4, which he has already proven and which we know as "Side Angle Side", to show that triangles AFC and AGB are congruent. Because of this, he can conclude that |GB| = |FC| and that the angles at F and G are congruent, and so he can again use Proposotion 4 and conclude that triangles BGC and CFB are congruent.

Since m∠FCA = m∠GBA and m∠FCB = m∠GBC, Euclid concludes that m∠ABC = m∠ACB.//

Isometries

In the proof above we used a sort of a "trick" -- we created a circle with center A through point F in order to find a point G that we needed. This "trick" comes up often in ruler and compass constructions, which we will learn about later.

The author of our textbook prefers to use a different set of tricks from the ones Euclid used. His preferred tools are isometries; isometries are flips or sliding motions that superimpose one object on another object of equal proportions. On page 13 the author gives a more precise definition of isometries as motions of the plane which preserve all distances between points in the plane.

The first proof we will look closely at says that if the side lengths of one triangle match the side lengths of another, those two triangles are congruent ("Side side side"). Since we're trying to take as little for granted as possible, we want to define the word "congruent" if we can.

In fact, we can use the idea of an isometry to define congruence! If we have one figure, and we can flip and slide it around until it exactly coincides with another figure, we say that those two figures are congruent. Two sets of points are congruent if there exists an isometry that maps one to the other.

I've talked a lot about "flipping and sliding", which takes many things for granted. The book is more careful with its fancy definition involving "distance". Axioms 6 through 8 on page 13 of the book describe the assumptions the author needs to make about isometries in order to use them the way he wants to. Axioms 6 through 8 also happen to describe different ways of flipping and sliding figures around in the plane.

Axiom 6: Given any points P and Q, there exists an isometry f so that f(P) = Q.

This just says that there's some way to slide (translate), turn (rotate) or flip (reflect) any point in the plane onto any other point.

Axiom 7: Given a point P and any two points Q and R which are equidistant from P, there exists an isometry which fixes P and sends Q to R.

Suppose you have three points arranged as shown:

  A


B   C
with |AB| = |AC|. Then you can either reflect (flip) across the vertical line through A to put B on to C, or you can rotate (spin) point B around point A until it lands on top of C. (This rotation would move C to a point above and to the right of where it is now -- the fact that isometries preserve distances between points means that C has to end up as far from B as it was before we moved B.)

In other words, you pick a point to stay still and a point to move around it to another point. There is a way to move the points like this (an isometry) only when the point you're moving to is the same distance away from the fixed (still) point as the point you started with.

Axiom 8: Given any line l, there exists an isometry which fixes every point in l but fixes no other points in the plane.

Reflecting (flipping) across a line is an isometry.

There are formal definitions of reflection, translation and rotation on page 16. (I believe that the author chooses isometries over the traditional tools from ruler and compass constructions because he prefers a transformational approach to teaching geometry. It's possible to teach high school geometry using either a classical or a transformational approach, depending on the materials available.)

Homework: Read page 15 and post a question about it on the message board.

Proof of "SSS"

Theorem 1.3.1: If the corresponding sides of two triangles ΔABC and ΔA'B'C' have equal lengths, then the triangles are congruent. How are we going to prove this? The definition of "congruent" tells us we have to find an isometry that maps one triangle to the other. It's too hard to do this all at once, so we do it step by step, finding an isometry for each step. (You'll convince yourself for homework that if you do one isometry then another, the resulting combination is also an isometry.)

Proof: It turns out that the proof the author uses doesn't work if the triangle is degenerate (all vertices are on the same line). He probably realized this after he finished proving it, but when he wrote down the final version of his proof he mentioned it at the beginning. The proof is left to the reader. In this case it's a fairly easy proof, but if you ask me in advance you might get a tiny bit of extra credit for writing it up and turning it in.

Here I'll go through the rest of the proof step by step, trying to relate make each isometry a separate step. Our plan is to prove that we can slide, flip and turn the plane until vertices A, B and C land on top of vertices A', B' and C', respectively. Step 1: Use isometry f1 to move A to A'.

Axiom 6 says there exists an isometry that does this. Let's call it f1. (Notice that all we need to know is that it exists; we don't have to know exactly what it is. This is common in mathematical proofs, and can lead to the odd situation of knowing something is possible but not how to do it.)

Step 2: Hold f1(A) at A' and move f1(B) to B'.

Here we use Axiom 7. We have a point A' = f1(A). We also have two points f1(B) and B' that are equidistant from A'. (We know these distances are equal because we started out knowing that |AB| = |A'B'|, and isometries don't change distances.) Axiom 7 says there exists an isometry that fixes (doesn't move) A' and sends f1(B) to B'. The author calls this isometry f2.

So f2(f1(A)) = A' and f2(f1(B)) = B'. If f2(f1(C)) = C' we're done. Otherwise, we need to:

Step 3: Move f2(f1(C)) to C' without changing anything on line A'B'. This step looks pretty easy. We have to leave f2(f1(A)) = A' and f2(f1(B)) = B' where they are and move f2(f1(A)) = A' and f2(f1(C)) to C'. Reflecting the whole plane through line A'B' ought to do that.

We know that such a reflection exists, and we know that that reflection will take f2(f1(C)) to a point that's distance |AC| from A' and to a point that's distance |BC| from B'. That describes point C', so that ought to finish the proof.

However we don't know that there isn't some sneaky point D that's also distance |AC| from A' and distance |BC| from B'. Lucky for us, the author proved in Chapter 9 that if two circles intersect, they intersect in at most two points.

We draw a circle around A' with radius |AC| and a circle around B' with radius |BC|. These two circles must intersect at points C and C'. Using the lemma the author proved in Chapter 9, we know the circles don't intersect at any other points, so there are no other points the right distances from A' and B'. Isometries preserve distances, and we can use Axiom 8 to find an isometry f3 that holds A' and B' in place while f2(f1(C)) moves, so that isometry must move f2(f1(C)) to C'. To summarize, we have proved that ΔABC and ΔA'B'C' are congruent by showing that there is an isometry f3 o f2 o f1 that maps A to A', B to B' and C to C'.//

Homework: Explain why a combination of two isometries (e.g. a translation followed by a rotation) is still an isometry.