I'm just going to assume I've constructed an equilateral triangle, so Triangle ABC is equilateral. and if it's equilateral, that means all of it's sides are equal, and lets say that it has an equilateral with sides of length x. so this is going to be x this is going to be x, and this is going to be x. we also know, based on what we seen from equilateral triangles before, that the measures of all of these angles are going to be 60 degrees.

We know that this is an equilateral triangle

So this is my best shot at an equilateral triangle.

And then the other wire is going to form an equilateral triangle.

You obviosly can't have a right triangle that is equilateral, because an equilateral triangle has all, all of their angles have to be 60 degrees.

Equilateral

Now lets start with a triangle that we're very familiar with... so let me draw ourselves an equilateral triangle... so drawing the triangles is always the hard part.

So I'm going to take this wire of length a and form an equilateral triangle out of it, right?

Equilateral Triangle

Triangle BEC is equilateral

The perimeter of an equilateral triangle ABC is 3 times the perimeter of an equilateral triangle DEF.

Equilateral, D.

It's an isosceles triangle

So this is my equilateral triangle.

An isosceles, let me write this, this is a right isosceles, i sos ce les triangle.

Equilateral triangle with given two vertices

So this is an equilateral triangle.

So if I had another isosceles triangle.

This is an upside down isosceles triangle, this is a right side up one.

Three wayward Pac Men, and three pointy brackets are actually present.

The remaining sides, I get a triangle each

This is an isosceles triangle, so the two legs are going to be the same.

This whole base of the equilateral triangle is 2a.

And this comes to a general formula for the area of an equilateral triangle.

So if you have a right isosceles triangle, whatever the two legs are, they're gonna have the same length, that's why it's isosceles.

So now, you know all three sides are equal, so this is an equilateral triangle.

Or another way to think about it is, if I have a right triangle that is also isosceles ,so a right triangle that is also isosceles.

If we have a right triangle and we have one of the sides (which ihappens to be congruent) and our hypotenuese is congruent.

So even thought, it doesn't look that way based on the way, its drawn. This is actually an isosceles triangle.

OK, so this is an equilateral triangle, all the sides are equal.

One is bent to form a square, and the other is bent to form an equilateral triangle.

Because these two base angles it's an isosceles triangle the two legs are the same so are two base angles

I can get another triangle out of these 2 sides of the actual hexagon

And it looks like I can get another triangle out of each of the remaining sides

And the only way it's going to be an isosceles triangle is if this length is equal to that length, so let's just try it.

And, if it's isosceles that means 2 of the sides are equal.

And then we if we drop an altitude, we know that this is we know that this is an equilateral triangle

I always forget the formula for the area of an equilateral triangle, so we'll have to kind of derive it right now.

Its kind of similar triangle where all of the sides are also of the same length

It is an equilateral triangle, at the base is a volcano between three castles, over which is a rainbow and the sun shining.

So the ratio of the sides and the hypotenuse, in a 45 45 90 triangle, or a right isosceles triangle.

So whatever isosceles triangle, if you have your 2 legs of it, the 2 base angles are going to be congruent

Well it's an equilateral triangle so all the sides are the same, so this is 2a and this is 2a.

So I'm going to make the equilateral triangle out of this side, and make the square out of the side, just because I have a sense it will make the math a little easier.

Notice, and I guess I didn't point this out that by dropping this altitude I've essentially split this equilateral triangle. into two 30 60 90 triangles.