A depressed cubic, followed by two special quadratics, and now a special rational function:

(1)

Everything about our triangle is *special*. For anyone without clue as to how the above rational function can be construed from a geometric point of view, our triangle is here to the rescue.

## Slipshod notation

Let be such that sides have slopes , as usual.

Denote the length of the altitude from vertex by and the length of the median from vertex by . We prove the following height-median-ratio:

(2)

## Simple deductions

#### Example

PROVE that .

We expect this to be the case — is the altitude from vertex , so it is a shorter distance compared to .

Alternatively, we can use the most basic form of the AGM inequality:

Suppose that the sides of have slopes , respectively. PROVE that the length of the altitude from vertex coincides with the length of the median from vertex if and only if .

Obviously, if , then the geometric progression is essentially . These slopes yield an isosceles triangle, from which the conclusion follows.

The advantage of having equation (2) is that it allows for “algebraic” argument. For example, recall that . If , then . That is, . And so . Conversely, if , then .

## Systematic derivation

Many of the nice properties of triangles with slopes in geometric progression () hinge on the following relations among the coordinates ():

(3)

In particular, what we establish below depend on these relations.

Let , , and be the vertices of in which sides have slopes . PROVE that .

Normally, . Using (3), we get:

Let , , and be the vertices of in which sides have slopes . Find the equation of the altitude from vertex .

The slope of the altitude from vertex is , since the slope of side is . Thus the equation of this altitude is

Using the expressions for and in equation (3), we obtain:

(4)

Let , , and be the vertices of in which sides have slopes . Find the *coordinates of the foot* of the altitude from vertex .

The equation of side is . The altitude from vertex has equation given by (4), namely:

Solving, we obtain

Let , , and be the vertices of in which sides have slopes . PROVE that *length* of the altitude from vertex is .

We find the distance from to the foot of the altitude

The -difference is

The -difference is

By the distance formula:

Let , , and be the vertices of in which sides have slopes . PROVE the *height-median-ratio*: .

We use example 3 and example 6 and part of our previous post:

to obtain

and then

## Sample applications

No doubt you already know the drill.

Find coordinates for the vertices of a in which the length of the altitude from vertex is the length of the median from vertex .

Without the organized procedure that equation (2) provides, solving this type of problem may be somewhat random. So as our custom is, we use what we’ve got. In equation (2), set

and solve the resulting quadratic for :

Fix . It now remains to find the first and third terms of a geometric progression with second term . For simplicity, let’s choose . Then the third term will be . The basic set of coordinates are:

Choose so that the slopes of are as per our set up. The length of the median from vertex is , and the length of the altitude from vertex is .

Find coordinates for the vertices of a in which the length of the altitude from vertex is the length of the median from vertex .

In equation (2), set

and solve the resulting quadratic for :

Fix . We now want a three-term geometric progression with as the second term. For simplicity, choose . Then we have as the progression. The corresponding coordinates, using the most basic form, will be , , and .

has coordinates at , , . Find the length of the altitude from vertex .

The side-slopes are for . They form a geometric progression in which and . The length of the altitude from vertex in such a case is given (see example 6) by

## Takeaway

Be c*ar*eful with the sign of in the rational expression:

because can be negative, whereas the ratio on the left is positive.

Note also that the “strange” formulas we state work only in the case of triangles with slopes in geometric progression. (By the way, it now seems that these triangles provide a platform for studying some basic functions.)

## Tasks

- PROVE that if the coordinates of a triangle’s vertices are related according to (3), then its side-slopes form a geometric progression.
- Let as in equation (1). For , PROVE that .
- Consider in which unit and units. If , PROVE that .
- Find coordinates for the vertices of a triangle in which the length of the altitude from vertex is the length of the median from vertex .
- (Explicit error) Find coordinates for the vertices of for which the length of the altitude from vertex is twice the length of the median from vertex .