How can one start with nothing, then end up with infinitely many things? Well, when one starts with a triangle whose slopes are not necessarily in geometric progression and then “enters inside” it; one obtains an assemblage of “smaller” triangles whose slopes are in geometric progression.

#### Desirable properties from sub-triangles

A sub-triangle is simply a (smaller) triangle contained in another triangle. It can share a side with the original triangle, its vertices can lie on the sides of the original triangle, or its vertices can lie completely inside the parent triangle.

The medial triangle of any triangle is a good example of a sub-triangle in which its vertices lie on the midpoints of the sides of the original triangle (for instance, in the diagram below, is the medial triangle of ).

A sub-triangle can also lie completely inside the parent triangle, like below:

We show that any triangle with one side parallel to the -axis contains infinitely many sub-triangles with slopes in geometric progression; in particular, we obtain that “coveted” case in which the common ratio .

#### Example (THEOREM)

Let be a right triangle in which one leg is on the -axis (or parallel to the -axis). For any real number , , PROVE that there’s a sub-triangle of whose side slopes form a geometric progression with common ratio .

To make the proof simple, consider a right triangle with vertices at . Every other right triangle that fits the description in our “theorem” is a translation of the named , so no loss in generality.

does the trick, if we define . The essence of requiring a positive is to ensure that lies __inside__ the given triangle.

Thus, the slope sequence is geometric, with common ratio .

Considering how simple and basic the above proof is, does the result deserve the use of that reserved — and almost revered — word “theorem”? Or, should the “label” be reversed?

#### Example

PROVE that each of the sub-triangles constructed in Example 1 above contains an obtuse angle.

Recall the slopes: . Since the common ratio is positive, it follows (depending on the sign of ) that the slopes of the sides are either all positive or all negative. Now, any triangle that contains all positive or all negative slopes must be obtuse-angled. (VERIFY this!)

Consequently, none of the sub-triangles constructed in Example 1 is equilateral or right-angled.

#### Example

For any right triangle with legs on the coordinate axes, PROVE that if the endpoints of the hypotenuse are joined to the centroid, then the slopes of the resulting sub-triangle form a geometric progression with common ratio .

COOL!

Since the centroid has coordinates , this is the case of in view of Example 1.

Note that the same result holds when the legs are parallel to the coordinate axes.

## Two times infinity

Not only do we get one sub-triangle for a given ; we actually get two — in fact, more!!!

#### Example

PROVE that every triangle with slopes contains a sub-triangle with slopes

We’ll apply the first proposition we established in our previous post; it states that the (non-zero) slopes of a triangle form a geometric progression if and only if there is a median whose slope is the negative of the slope of the side it meets.

Suppose that is such that sides have slopes . Then the slope of the median from vertex is :

Let be the midpoint of and let be the midpoint of . Join , the dashed red line segment shown above. Since connects the midpoints of two sides, it is parallel to the third side, namely . So, the slope of is equal to the slope of , which is . Furthermore, the slope of is , same as that of .

Thus, has slopes for sides , respectively.

Note that there are actually two sub-triangles with slopes associated to a parent triangle with slopes .

#### Example (THEOREM)

Let be a non-right triangle in which one side is on the -axis (or parallel to the -axis). For any real number , , PROVE that there’s a sub-triangle of whose side slopes form a geometric progression with common ratio .

This follows from Example 1. Since one side is parallel to the -axis, draw an altitude from the opposite vertex. The two resulting right triangles both conform to the description in Example 1.

#### Example (Main goal)

PROVE that every triangle with one side parallel to the -axis contains a sub-triangle in which one median is vertical and one median is horizontal.

The proof of Example 6 follows from Example 3 and Example 4.

EUREKA! Remember that “VHM” property? See here. Since the “VHM” property is extremely nice, we were curious as to whether it can be embedded into every triangle. Today’s post arose from that curiosity. We’ve now done it for triangles in which one side has zero slope; we’ll do something similar for general triangles.

## Three times infinity

We improve on the number of sub-triangles we can have for any given .

#### Example (Theorem)

PROVE that any right triangle with legs parallel to the coordinate axes contains three different sub-triangles whose slopes form a geometric progression with common ratio , where .

We prove this for the right triangle with coordinates ; just a slight modification is needed for the general case when the legs are parallel to, but not on, the coordinate axes.

The desired sub-triangles, each with side slopes in geometric progression and common ratio , are: . Using the given coordinates, we can confirm this by calculating slopes:

Thus, the slopes of the sides of differ by a constant multiple of . Since we’ve already shown that the slopes of differ by a constant multiple of , it just remains to do the same for .

Therefore, the slopes of the sides of form a geometric progression with common ratio .

Observe that in the above diagram has side slopes also in geometric progression, but with common ratio .

#### Example (zig-zag theorem)

For any right triangle with legs () on the coordinate axes, PROVE that there are points and on the hypotenuse and an interior point such that the slopes of form a five-term geometric progression.

Let the vertices of the right triangle be located at . For , define . Then is an interior point of ; further, and lie (internally) on the hypotenuse . With the given coordinates, we have the following slopes:

These slopes differ by a constant multiple of . Notice how the line segments are traversed in a sort of “zig-zag” manner as shown below:

## Numerical problems

Always a good idea to support theory with examples.

#### Example

Given with vertices at , find coordinates for its four sub-triangles that all satisfy the “VHM” property.

Note that the parent triangle doesn’t satisfy the “VHM” property; in particular, the -coordinates do not form an arithmetic progression, though the -coordinates do.

A sample sub-triangle is with vertices shown above. Notice that its -coordinates form an arithmetic progression, as well as its -coordinates .

How did we obtain ? Easy. We followed Example 3 and Example 4:

- divide the parent triangle into two right triangles ( and )
- find the centroids of each of the two resulting right triangles (for it is )
- connect the centroid to the endpoints of the hypotenuse (e.g., for , join to and to ; this way, the resulting triangle has common ratio )
- connect the centroid to the midpoint of the hypotenuse ( in the diagram)
- join to the midpoint of ( in the diagram). DONE.

The centroid (), the midpoint of the hypotenuse (), and the midpoint of side () are all we need. Repeating this process for the lower sub-triangle as well as for the right triangle on the left, we obtain the following list of four sub-triangles satisfying that satisfying “VHM” property (you read that correctly):

- ;
- ;
- ;
- .

Of course, each is a peach.

#### Example

Given the right triangle with coordinates , find coordinates for two points and on and an interior point such that there is a five-term geometric progression for the slopes of .

Set . Then we have:

Thus the slopes of the segments are , respectively. They form a five-term geometric progression with a common ratio of .

Having read through this page, it’s now too late to hate these greats — triangles with slopes in geometric progression.

## Takeaway

Sometimes, you already have what you seek — if you look “within”. Does that seem to click?

As we’ve shown, a parent triangle may not have a desirable property on the surface, until its interior is examined. To look down on people — or things — solely on their outward look, is not cool.

## Tasks

- (Embedded hexagon) For any triangle , PROVE that it is possible to embed a hexagon whose vertices lie completely inside , and such that three of its sides are the side lengths of and the remaining three are the side lengths of .
(In the diagram above, . Simply find the coordinates of in terms of the coordinates of .)

- (Power two) Given with vertices , the slopes of sides form a geometric progression with common ratio . Find coordinates for two points and on such that the slopes of the sides of form a geometric progression with common ratio .

(Notice that . This is always possible for any triangle whose slopes form a geometric progression with a positive common ratio.) - (Power two) Given with vertices , the slopes of sides form a geometric progression with common ratio . Find coordinates for two points and on such that the slopes of the sides of form a geometric progression with common ratio .

(Notice that . This is always possible for any triangle whose slopes form a geometric progression with a positive common ratio.) - (Five terms) For any triangle with slopes of sides forming a geometric progression with positive common ratio, PROVE that there are points and on such that the slopes of the line segments form a five-term geometric progression.

(This has very important implication and application. In particular, it helps us to associate every finite geometric sequence having a positive common ratio with a triangle.) - Given with vertices , its slopes () are in geometric progression. PROVE that it contains a sub-triangle satisfying the following two properties:
- its slopes are in arithmetic progression, namely
- the midpoint of one of its sides is the centroid of the original triangle.

(The above triangle has very nice properties; no wonder we’ve encountered it twice in these tasks.)

- (Two sequences) Let be such that the slopes of sides form a geometric progression with a positive common ratio . PROVE that there is a point on , a point on , and a point on such that the slopes of the sides of form an arithmetic progression .
- Let be such that the slopes of sides form a geometric progression, with a
__positive__common ratio. PROVE that there are points and on such that . - Let be such that the slopes of sides form a geometric progression, with a
__negative__common ratio. PROVE that there is a point on and a point on such that .

(Apparently, negative common ratios change things slightly. Being biased, they are our favorite.) - (Internal division) Let be a line segment, where has coordinates and has coordinates . For , PROVE that the point lies
__within__. - (Smaller lengths) PROVE that every triangle contains a sub-triangle whose lengths are of the lengths of the original triangle, for any .