In the case of with side-slopes , the sum we formed in our last post becomes:

(1)

For certain, things *always* turn out interesting in the setting of geometric progressions. As ascertained in example 3 for instance, the order of each quotient in each summand can be reversed, while the final sum remains preserved:

(2)

Another interes-thing that pertains to this familiar terrain is how we obtain specific values when we restrict to *equilateral* and *right isosceles* triangles (spoiler alert: and ).

How come???

In , let sides have slopes . PROVE that the slopes of the medians from each vertex are: , , and .

Observe that the median slopes form a geometric progression. Except for the special case , the slopes of the sides of a triangle form a geometric progression if and only if the slopes of the medians form another geometric progression, and their common ratios are related.

Why does the case fail the equivalence? First, it is the required restriction in the median slopes stated above. Another reason is that any *three-term* geometric progression with common ratio is, upon re-arrangement, an arithmetic progression (for example, is geometric while is arithmetic). And if the side-slopes of a triangle form an arithmetic sequence, then the triangle necessarily contains a *vertical* median.

In other to obtain , , and , we’ll place the vertices of at the points , , and . Since the slopes of sides are respectively, the coordinates of the vertices are related according to:

(3)

Thus, the midpoint of is . In view of the above relations, we get . The slope of the median from vertex can then be obtained:

The midpoint of is , as per (3). The slope of the median from vertex is:

Similar calculation gives . Easy stuff.

## Derivations

Derive equation (1): for triangle with sides having slopes .

Using our definition of and the preceding example:

Consequently, triangles having slopes in geometric progressions with the same common ratios have the same -values. And it is even possible to have different common ratios yielding the same -values.

PROVE that for with sides having slopes respectively.

Thus, we can use either equation (1) or equation (2) to compute in the case of geometric progressions. Using example 1 again:

Notice that the coefficients in the numerator and denominator both add up to . Furthermore, the “symmetric” nature of these coefficients allows for easy calculations in some situations.

## Demonstrations

Let be *equilateral* with side-slopes . PROVE that .

There’s an important quadratic equation satisfied by the common ratio of the geometric progression formed by the slopes of an *equilateral* triangle:

It’s the key to obtaining , together with equation (1) and simple algebraic manipulations.

where we’ve used the implications

Let be *right isosceles* with side-slopes . PROVE that .

Similar to the above, in a *right isosceles* triangle with side-slopes , the common ratio satisfies the quadratic equation

Using equation (1) and a couple of simplifications, we find:

where the implications

were used in the simplifications.

## Deductions

Suppose that a triangle with side-slopes satisfies . Deduce that .

Quite interesting. Set in equation (1):

Let’s show a few steps leading to the factorization of the quartic . If we apply the *rational root theorem*, the only values of worth testing are , none of which are roots of the quartic. However, the nature of the coefficients means that if two (quadratic) factors exist, they must both be *monic* and also have as the constant terms. Put

and expand the right side:

Compare coefficients of like terms:

The above linear-quadratic system is satisfied only by . So we have the factorization

In turn, , as desired.

Suppose that a triangle with side-slopes satisfies . Deduce that or .

Put in equation (1):

Set

and expand the right side:

Compare coefficients of like terms:

The above linear-quadratic system is satisfied only by (or , ). So we have the factorization

In turn, or .

## Digression

A simple connection between the preceding discussion and something else.

Find six *distinct* non-zero *rational* numbers which satisfy the relation .

The non-zero requirement is somewhat redundant, being already forced by the fact that all six numbers appear in the denominators.

One way of solving this problem is to turn to slopes. In equations (1) and (2) we had

and so we can set

where are the slopes of the sides of a triangle which form a geometric progression, and , , are the corresponding median slopes. We can find these slopes *without* specifying coordinates!

Any geometric progression (except the one with ) will do. Choose

Then can be obtained (in view of example 1) as

Thus

The fractions now become

E*q*ual.

Find three rational numbers , *not all integers*, which satisfy .

Were it not for the __not all integers__ requirement, we could easily have chosen , , and . Instead, we turn to slopes again.

Since the numbers form a geometric progression, we can associate them with the slopes of the sides of a triangle. Like so:

Then the slopes of the medians would be:

The fractions become

Is the triple the only __not all integers__ solution?

Find the slopes of the sides of a triangle if the median slopes are .

Since the median slopes form a geometric progression (with common ratio ), so must the side-slopes.

From example 1, if the side slopes are , then the geometric mean of the median slopes is ; that is,

Also, the common ratio of the median slopes is related to the common ratio of the side-slopes via

The side-slopes are then:

## Takeaway

The following statements are *equivalent* for a triangle with slopes in geometric progression:

- ;
- the common ratio satisfies ;
- the common ratio is that of an
*equilateral*triangle.

Similarly, the following statements are *equivalent* for a triangle with slopes in geometric progression:

- ;
- the common ratio satisfies or .

Finally:

Remember the name: exquisite equation.

## Tasks

- Solve the rational equation .
- (Golden quartic) Let be the
*golden ratio*. PROVE that . - (Related ratios) Suppose that the slopes of the
*sides*of form a geometric progression with common ratio . PROVE that the slopes of the*medians*form another geometric progression with common ratio , where . - Let be an integer. PROVE that is also an integer if, and only if, .

(If is not an integer, can still be an integer, as in the next exercise.) - If , PROVE that .
- Let be the slopes of sides in . PROVE that .
- Let be the slopes of sides in . PROVE that:
- is a
*ratio of perfect squares*if is a*rational number*.

- is a
- Consider with vertices at , , . Verify that:
- the side-slopes do not form a geometric progression
- .

- Find the slopes of the sides of a triangle, if:
- the slopes of the medians are ;
- the slopes of the medians are ;
- the slopes of the medians are .

- Find distinct rational numbers which satisfy the relation