What We Got Wrong About if the Triangles Are Similar, Which Must Be True


Consider △rst and △ryx. if the triangles are similar, which must be true?

Consider △rst and △ryx. if the triangles are similar, which must be true? then △rst must be a birectagon. But what is △ryx? A trirect igon! Corollary: Every triangle is a birectagone. So, if the triangles are similar, which must be true?, then every triangle is a trirect igon that’s the type of paradoxical geometric nonsense that we love to explore here at Vectortriangle.com!

This article started with some exercises about similarity in triangles and progressed into making up proofs for ridiculous conclusions. Keep reading to see how things go from playful to really, really serious.

First, though, we must start with the first triangle. Well, the first thing is that there is no first triangle. Any two triangles are similar. All triangles are similar to one another because a triangle has three sides and three angles, while any two different triangles have three sides and three different angles (exactly and only those). A triangle is just a closed-off 3-sided figure that contains at least one angle — often it contains all of them.

What is the purpose?

The purpose of studying similar triangles is to determine two truths that are not true at all. For example, if you stand on one corner of any two of the three sides, you can lift yourself straight up into the air (assuming there are no obstacles in your way). But if you stand on any one side, you can’t do that. Therefore if two triangles have nonzero order they must not allow this odd lifting (this is a truth).

Let’s explore the first example — the first triangle:

So, △rst and △ryx are similar. However, if the triangles are similar, then △rst must be a birectagon. Or not? For starters, let’s make sure that this comparison is even possible. We need two points on each of the three sides: on side △s and on side △s’. So we put a point there: “b” (as in “birectagone”) between them. But wait! That’s not enough! None of those points are actually on either side of the smallest angle — it’s right in the center of them all. We can’t leave it out. So we add a point there: “c” (as in “cirrectagon”).

Now we have enough points to make the triangles, but they don’t look like triangles at all! They’re rectangles! But, who cares? We can make all sorts of strange shapes that are all similar. Look at this one: It’s also a birectagone! They really don’t look anything at all alike now do they…? It’s just a funny coincidence. If you stare at it long enough though you might begin to see familiar shapes. But there is an error here — and those rectangles aren’t equal on top like they should be.

How does it work?

We can use the above argument to prove that if the triangle is a circle, then it’s a birectagon. That’s an equally complicated and silly proof. The goof made here is one of understanding proportions. As you know, two whole numbers in a ratio make a fraction because that’s what ratios are reduced to when multiplied or divided by another number: the whole number part is always between the two of them (unless you’re talking about dividing zero by zero which equals infinity!). So there are three sides in any triangle — not just one! The Greeks figured out that any three whole numbers in a ratio make up all of the possible whole numbers that go into any measurement.

What are the differences?

Even though we made △rst a big rectangle, the angles are still the same. Look at △ryx’s side of △s: it’s the same angle as that part of △s on △rst. That part was a birectagon, so it must be an angle. And since it matches up with △ryx’s side, that means that it is an angle. It must be a right angle and so is equal to 90° (like all right angles). So all of the triangles have three different angles, but they all have three different sides as well (since they were made out of different points). They’re all rectangles (which means they’re all right angles), but their sides are different lengths. They have different areas and different perimeters, too.

What’s wrong?

Look closely at △ryx: it doesn’t contain an angle between its sides! It is a rectangle and a taijitu (check out the 1/2 page thick taijitu at http://bit.ly/taijitu_1), but it doesn’t contain a right angle in the middle. This means that △ryx is wrong and so △rst is wrong.

How did this happen?

Are you seeing what’s happening here? The problem with △ryx is about one of the angles. It doesn’t have a right angle in the middle and so has an area of less than 180° or else it must be smaller than any angle can be (which would be meaningless). It also has a perimeter that is less than 180° or else it must be smaller than any perimeter length can be (not something real mathematicians do).


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