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# A note on Kepler triangle

A Kepler triangle is first of all a right triangle, and so it enjoys the Pythagorean identity:

(1)

In addition, it has the property that its side-lengths form a geometric progression, for example:

(2)

Equations (1) and (2) force the ratio of the hypotenuse to one of the legs ( or ) to be golden:

(3)

In this post, we show that a Kepler triangle also satisfies

(4)

which, in conjunction with equation (1), constitutes what one may call a double Pythagorean identity. We also show that if any of the four equations above hold, then the remaining three become equivalent.

Show that the Kepler triangle satisfies equation (4).

For the Kepler triangle we have , where is the golden ratio. By definition of , we have and so .

Note that a random right triangle need not satisfy equation (4). For example, the very familiar case with , , doesn’t.

## Simple characterizations

A couple of trivial equivalent statements that offer alternate descriptions of the Kepler triangle.

Suppose that a triangle satisfies equation (1). PROVE that equations (2), (3), (4) become equivalent.

In other words, the following three statements are equivalent for any right triangle (in which ):

1. the side-lengths form a geometric progression
2. the ratio is the golden ratio
3. the identity holds.

The result follows if we show that .

So let’s suppose that holds, that is, the sequence is geometric. This gives . Since we also have that , it means that

Thus, satisfies the defining equation for the golden ratio, and so holds.

Next, we show that . So suppose that is the golden ratio. Then

Multiply both sides of by to get

since from the Pythagorean identity and also from the golden ratio. Thus, holds.

Finally, we show that . We have and :

So the sequence is geometric, and is proved.

Easy equivalence.

Suppose that a triangle satisfies equation (2). PROVE that equations (1), (3), (4) become equivalent.

In other words, the following three statements are equivalent for any triangle in which :

1. the triangle is a right triangle with
2. the ratio is the golden ratio
3. the identity holds.

Follow the same argument as in example 2, bearing in mind that holds throughout in this case.

Suppose that a triangle satisfies equation (3). PROVE that equations (1), (2), (4) become equivalent.

In other words, the following three statements are equivalent for any triangle in which :

1. the triangle is a right triangle with
2. the side-lengths form a geometric progression
3. the identity holds.

Follow the same argument as in example 2, bearing in mind that holds throughout in this case.

Suppose that a triangle satisfies equation (4). PROVE that equations (1), (2), (3) become equivalent.

In other words, the following three statements are equivalent for any triangle in which :

1. the triangle is a right triangle with
2. the side-lengths form a geometric progression
3. the ratio is the golden ratio .

Follow the same argument as in example 2, bearing in mind that holds throughout in this case.

## Similar concept

Given in which , the Pythagorean theorem states that

(5)

(Or the usual .) But then the left side of equation (5) is just the sum of the squares of two altitudes — the altitude from vertex and the altitude from vertex — and so we can re-write (5) as:

(6)

where and are the altitudes from and . The latter equation expresses the fact that in a right triangle, the sum of the squares of two altitudes is equal to the square of the third side (the hypotenuse). We’ll use this intrinsic characteristic of right triangles as the defining condition for the triangles under consideration in the second part of today’s discussion.

PROVE that if, and only if, , where is the circumradius of triangle .

First suppose that . Since and , we get

Conversely, suppose that . Then:

QED:

PROVE that or , if .

Begin with as per the extended law of sines. The rest is:

PROVE that , if a non-right triangle satisfies .

Since the given triangle is not a right triangle, we take from the preceding example. However, since is always equal to in any triangle, we have:

Similar to equation (4) for the Kepler triangle. Let’s take a shot at making it exact.

Consider a non-right triangle that satisfies . PROVE that if, and only if, is the golden ratio.

First suppose that . Since from the preceding example we must then have . Positive square roots: , or , and so is the golden ratio. The converse is also easy. So basically, we have non-right triangles that tend to behave like the Kepler triangle.

## Special case

A sample triangle that satisfies equation (6) twice.

Consider an isosceles triangle in which , , and

For this triangle we have:

## Takeaway

For any non-right triangle with side-lengths , circumradius , and altitudes , the five statements below are equivalent:

In addition, if a non-right triangle satisfies any of the equivalent conditions above, then each of the following three statements implies the others:

• is the golden ratio
• is a geometric sequence.

In a nutshell, we have a non-right triangle version of the Kepler triangle.