## Extension confirmed

In one line of equations, the extension that arises from a combination of (1) and (2) is:

(3)

Please don’t take equation (3) too seriously: there are many relations in Euclidean geometry that can be expressed in terms of the circumradius of a triangle, and equation (3) is just one of them. Meanwhile, note that:

- one of the cosine terms in (2) (or (3)) requires an absolute value if the parent triangle is obtuse
- only right triangles fail some of the equalities in (2) (or (3)).

*acute*triangle with circumcenter . Denote by the circumradii of the triangles , in that order. Verify equation (2), namely: , where is the circumradius of the parent triangle .

Whether the parent triangle is acute or obtuse, at least one of the associated triangles has to be obtuse. However, this won’t affect the calculations since all we have to do is use the extended sine law.

Here is acute, so it’s circumcenter is located inside the triangle as shown above. Consider . By the inscribed angle theorem, we have:

Now apply the extended sine law to triangle , noting that :

and so . In the case where one of the triangles is obtuse, we also obtain the same result.

*obtuse*triangle with circumcenter . Denote by the circumradii of the triangles , in that order. Verify equation (2), namely: , where is the circumradius of the parent triangle .

In this case, the circumcenter is located outside the triangle. A sample diagram is shown below:

In , angle is equal to , while angles and are both equal to . Apply the extended sine law again to get:

This is where an absolute value is needed.

## Exception circumvented

If the parent triangle is right-angled, let’s say at , then and so we need modify equation (2) slightly.

We would have and . Using equation (2) we have:

This gives .

Note that since triangle is right-angled at , we have . From isolate :

Similarly, . Together we get . And

## Exploration continues

We’ll have need to use equation (2) in future. For now, consider what happens in the isosceles triangle.

## Equation changes

An update about our ongoing *hide-and-seek* game with some hackers. As you can see below, we were initially hoping to use an exponential equation to model their menace, but the data suddenly changed after October 28, and so did our equation.

How wonderful would it be if the hackers spent all that time focusing on analytic geometry?

## Takeaway

In an *obtuse* triangle , let be as mentioned in the post. Then the following statements are *equivalent*:

- .

If instead triangle is *acute*, then the following statements are *equivalent*:

- .

## Tasks

- (Similar triangles) Let be a
*non-right*triangle with circumcenter . Denote by the circumcenters of the triangles , in that order.- PROVE that is
*similar to the orthic triangle* - Find the similarity ratio.

- PROVE that is
- (Similar triangles) Let be a
*right*triangle with circumcenter and . Denote by the circumcenters of the triangles , in that order.- PROVE that is
*similar to the parent triangle* - Find the similarity ratio.

- PROVE that is