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# Thanksgiving Theorems II

It’s becoming our tradition to have special editions on Thanksgiving occasions, and this year is no exception. Thus, there’s a brief digression from our familiar and friendly and favourite geometric progression, to arithmetic progression.

## Golden ratio

The main equation for today’s consideration is

(1)

There’ll be explanation for it in examples 9 and 10. For now, look what we get if we set :

a.k.a golden ratio. You’ve seen this before here, and you’ll meet it again before the end of this year.

## Gentle reminder

Kindly remember that the due date to hand in your solutions to examples 1 through 8 is October 28. Please don’t hesitate, don’t leave it late.

Find the circumcenter of with vertices at , , .

It is located at .

Find the orthocenter of with vertices at , , .

It is located at .

For with vertices at , , , find the reflection of the orthocenter along side .

It is the point , and it lies on the circumcircle of .

For with vertices at , , , PROVE that the point is on the circumcircle of .

We use examples 1 and 3. Specifically, the midpoint of and is precisely the circumcenter .

Since is on the circumcircle and the midpoint of and is the center of this circle, it follows that is also on the circumcircle.

For with vertices at , , , PROVE that the point and the point are end points of a diameter of the circumcircle of .

Use example 4 above.

Find the orthocenter of with vertices at , , .

Side is parallel to the -axis this time around. The orthocenter is located at .

Find the reflection, along side , of the orthocenter of with vertices at , , .

It is the point .

Find the circumcenter of with vertices at , , .

It is the point .

## Geometric result

#### (First “Theorem”)

In , let and be the slopes of sides and , with side parallel to the -axis. Let be the orthocenter and let be the reflection of the orthocenter along side . Then there is a point on the circumcircle of such that the slopes of the medians of form an arithmetic progression with common difference , where .

As you traverse the sides of you get familiar terms: Human Resources, Registered Nurse, Northern Hemisphere. You’ll never forget .

For convenience, place the vertices of at the points , , . Let be as given in examples 2,3,4:

The midpoints of sides are:

respectively. And so the slopes of the medians from are:

We get an arithmetic progression with common difference

Returning to the parent triangle , we have

Then, then, and then

#### (Second “Theorem”)

In , let be the slopes of sides , with side parallel to the -axis. Let be the orthocenter and let be the reflection of the orthocenter along side . Then there is a point on the circumcircle of such that the reciprocals of the slopes of the medians of form an arithmetic progression with common difference , where .

For convenience, place the vertices at , , . Take the orthocenter , its reflection along side , and the point . The mipoints of sides are:

respectively. The slopes of the medians are then

and their reciprocals (re-arranged) are

These form an arithmetic sequence with common difference , where

In terms of the parent triangle (vertices at , , ), we get:

## Takeaway

The construction we’ve done is true more generally, as follows:

• If a triangle has one side parallel to the -axis, then the reciprocals of the slopes of the three medians form an arithmetic progression;
• If a triangle has one side parallel to the -axis, then the slopes of the three medians form an arithmetic progression.

1. Suppose that the enumeration represents an arithmetic progression of distinct, non-zero numbers.
• if is also arithmetic, PROVE that .
• if is also arithmetic, PROVE that .
• if is also arithmetic, PROVE that .
2. Find three distinct, non-zero numbers in arithmetic progression for which the reciprocals are also in arithmetic progression.
3. Find three distinct, non-zero numbers in arithmetic progression for which the reciprocals are also in arithmetic progression.
4. (Registered Nurse) For with vertices at , , and , PROVE that the points and are end points of a diameter of the circumcircle of .
5. Find coordinates for the vertices of a triangle in which the slopes of the three medians are .

## Thanksgiving

Nationally, Thanksgiving is just one day — this past Monday; naturally, ours lasts more days, many days — being our mainstay.

Our style of incorporating Thanksgiving into an “academic setting” is not very popular, and you may not be familiar with it. It’s part of what makes us peculiar; and in particular, it’s due to a spectacular event on Thursday, June 14, 2018: beautiful, remarkable, unforgettable day. Consequently (and continuously and conspicuously), the poster expresses gratitude to the Ancient of Days.

All things being equal, our next iteration of Thanksgiving comes up on Monday, June 14, 2021. Until then, please understand that this practice is very useful for us, as we’re hopeful that we’ll always be mindful of the need to be grateful.