How to Construct a Brocard Circle

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will explain a Brocard Circle and look at the steps of how to construct a one using its characteristics, a straight-edge, and a compass. Lastly, we will see an example to illustrate this process.

Brocard Circle

It's vacation time! Suppose that a fancy hotel resort, Mathematica, has a circular pool area, with a triangular pool. There is a circular food and drink bar, named the Brocard Bar, within the triangular pool as shown in the image.


brocardcons1


Hmmm…the Brocard Bar is an interesting name for a bar! You may be wondering where it came from!

It turns out that the circular bar is actually an important mathematical characteristic of the triangle that is the pool in this scenario, called a Brocard Circle. To define the Brocard Circle of a triangle, we need to be familiar with a few other definitions. Those are as follows:

  • Circumcircle of a triangle: The circle that passes through each of the vertices of the triangle.
  • Circumcenter of a triangle: The center point of the circumcircle of the triangle.
  • Median of a triangle: A line from a vertex to the center of the side opposite that vertex.
  • Angle bisector of a triangle: A line segment that splits a vertex's angle in half.
  • Symmedians of a triangle: The line segments formed by drawing a median of a triangle, and reflecting it over that vertex's corresponding angle bisector.
  • Symmedian point of a triangle: The intersection point of the three symmedians of a triangle.


Diagram of Definitions
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Phew! That's a lot of definitions, but now we have what we need to define the Brocard Circle of a triangle.

  • Brocard Circle of a triangle: The circle that has a diameter that is the line segment connecting the circumcenter of the triangle and the symmedian point of the triangle.


Brocard Circle
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Ah-ha! Now the bar's name makes complete sense! The bar is the Brocard Circle of the triangular pool!

How to Construct a Brocard Circle

When the Brocard Bar was being created, the architects and engineers that created the blueprints for the design had to be able to construct the Brocard Circle of the triangular pool. As it turns out, if one is familiar with a few simple compass and straight-edge constructions and can find the circumcenter and symmedian point of a triangle, this process is not too difficult. The steps are as follows:

  1. Find the circumcenter of the triangle.
  2. Draw in the symmedians of the triangle and find the symmedian point of the triangle.
  3. Use a straight edge to connect the circumcenter and the symmedian point of the triangle with a line segment. This is your diameter of your circle, called the Brocard Diameter.
  4. Find the midpoint of the line segment you just created.
  5. Use a compass to create a circle that has the midpoint you just found as its center and the symmedian point and the circumcenter as points on the circle.

Example

Okay, let's construct the Brocard Circle of the triangle RST shown.


Triangle RST
brocardcons9


Step 1 is to find the circumcenter of the triangle, so we draw in the circumcircle and find its center.


Step One
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Step 2 is to draw in the symmedians of the triangle and find their intersection point, or the symmedian point.


Step Two
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