Brocard points

Brocard points are special points in the triangle; named after the French mathematician Henri Brocard ( 1845-1922 ). Brocard was best known for the following sentence:

In a triangle with sides there is exactly one point such that the distances of the order form the same angle with the sides, ie that the angular equation. This point is called the first Brocard point and the angle is called the Brocard angle of the triangle.

There is a second Brocard point of triangle ABC; this is the point Q for the routes AQ, BQ, CQ include the same sequence to the sides b, c, a angle, ie applies. Oddly enough, this corresponds to the second Brocard point is the same Brocard angle as the first Brocard point, ie, the angle is the angle the same.

The two Brocard points are closely related; in fact, the difference from the first to the second depends on the order in which one takes the corners of the triangle ABC! For example, the first point of the triangle ABC Brocard is also the second point of the triangle Brocard ACB.

Before Brocard being studied already by August Leopold Crelle (1817 ) and Karl Friedrich Andreas Jacobi ( 1825).

Construction

The elegant design of the Brocard points, illustrated by the example of the first Brocard point below goes as follows (see figure):

Cutting the perpendicular bisector of the side AB with the perpendicular to the straight line through the point BC as the intersection to draw a circle so that it passes through the point B. Then this circle also passes through the point A and touches the line BC at point B. Similarly, we construct a circle through the points C and B, which touches the straight line CA at point C, and a circle through the points A and C, the the line AB at point A touches. These three circles have a common point - the first Brocard point of triangle ABC!

The three channels are constructed precisely referred to as the triangle ABC Beikreise. Analogously one constructs the second Brocard point.

Formulas for the Brocard angle

If you write for the area of ​​the triangle ABC, so can the Brocard angle with the following formulas:

• .

For each triangle.

Properties

• The two Brocard points of a triangle ABC are always isogonal conjugate to each other.
• The center of the two Brocard points ( Kimberling Number X ( 39) ) lies on the so-called Brocard axis connecting the circumcenter and the Lemoine point. The straight line joining the two Brocard points is perpendicular to the Brocard axis.

Coordinates

Third Brocard point

Occasionally, the point is called a "third" Brocard point. He has the Kimberling number and the barycentric coordinates, so that it closes the loop with the first two Brocard points with the coordinates or baryzentischen.