## Length of a Brocard Diameter

Since the bridge is across water, the builder is having a hard time determining how long the bridge needs to be. Thankfully, we have a couple of formulas that will help him out! You see, we have a nice formula for the length of the Brocard diameter of a triangle based on the side lengths of the triangle. This formula is also based on the length of the circumradius of the triangle, so first, let's look at the formula for the length of the circumradius of a triangle.

Now that we know the formula for the length of the circumradius of a triangle with side lengths *a*, *b*, and *c*, we can learn the formula for the length of the Brocard diameter of a triangle with side lengths *a*, *b*, *c*, and circumradius of length *R*.

Great! So to find the length of the Brocard diameter of a triangle with side lengths *a*, *b*, and *c*, we use the following steps:

- Plug
*a*, *b*, and *c* into the formula for the circumradius, *R*, of the triangle, and calculate the circumradius.
- Plug
*a*, *b*, *c*, and *R* into the formula for the length of the Brocard diameter of the triangle and simplify to get the Brocard diameter.

Let's give it a go with finding the length of this bridge so that the builder can get to work!

## Example

To find the length of the bridge, we are finding the length of the Brocard diameter of the triangular plot of land, with side lengths 3 kilometers, 4 kilometers, and 5 kilometers, so we can just take it through our steps.

The first step is to plug the side lengths *a* = 3, *b* = 4 and *c* = 5 into the formula for the circumradius of a triangle.

That's not as bad as we thought it might be. By simply plugging in some values and simplifying, we get the circumradius of the triangular plot of land to be 2.5 kilometers.

On to the next, and final, step! We just plug *a* = 3, *b* = 4, *c* = 5, and *R* = 2.5 into the formula for the Brocard diameter of a triangle and simplify.

We did it, and it was actually fairly easy with these formulas. We have that the length of the Brocard diameter, and hence the length of the bridge, is approximately 1.4 kilometers.

## Lesson Summary

To discuss the Brocard diameter of a triangle, we must be familiar with many different parts of the triangle, such as the medians, angle bisectors, symmedians, symmedian point, circumcircle, circumradius, circumcircle, and the Brocard circle.

The **Brocard diameter** of a triangle is the diameter of the triangle's Brocard circle that runs from the symmedian point of the triangle to the circumcenter of the triangle. We can calculate the length of the Brocard diameter of a triangle with side lengths *a*, *b*, and *c*, by first calculating the length of the circumradius, *R*, of the triangle using a formula, and then plugging *a*, *b*, *c*, and *R* into the formula for the length of the Brocard diameter.

As we've seen, these formulas make things fairly simple when trying to find the length of a triangle's Brocard diameter. Since all triangles have a Brocard diameter, and triangles show up often in the real world, it is a good idea to keep this information tucked in our back math pocket in case we need it in the future!