Brocard points

Brocard points are special points in the triangle ; named after the French mathematician Henri Brocard (1845–1922).

definition

Brocard was best known for the following phrase:

The first Brocard point P

In a triangle with the sides there is exactly one point such that the lines in turn enclose the same angle with the sides , i.e. that is, the angle equation holds. This point is called the first Brocard point and the angle is called the Brocard angle of the triangle . ${\ displaystyle \ Delta ABC}$ ${\ displaystyle a, b, c}$ ${\ displaystyle P}$ ${\ displaystyle {\ overline {AP}}, {\ overline {BP}}, {\ overline {CP}}}$ ${\ displaystyle c, a, b}$ ${\ displaystyle \ omega}$ ${\ displaystyle \ angle PBC = \ angle PCA = \ angle PAB}$ ${\ displaystyle P}$ ${\ displaystyle \ omega}$ ${\ displaystyle \ Delta ABC}$

There is a second Brocard point of the triangle ABC; that is the point Q for which the segments AQ , BQ , CQ in turn enclose the same angles with the sides b , c , a , ie for which applies. Strangely enough, this second Brocard point corresponds to the same Brocard angle as the first Brocard point, ie the angle is equal to the angle . ${\ displaystyle \ angle QCB = \ angle QBA = \ angle QAC}$ ${\ displaystyle \ angle PBC = \ angle PCA = \ angle PAB}$ ${\ displaystyle \ angle QCB = \ angle QBA = \ angle QAC}$

The two Brocard points are closely related; in fact, the distinction between the first and the second depends on the order in which the corners of triangle ABC are taken! So is z. B. the first Brocard point of the triangle ABC at the same time the second Brocard point of the triangle ACB.

Before Brocard, they were examined by August Leopold Crelle (1817) and Karl Friedrich Andreas Jacobi (1825).

construction

Construction of the first (P) and the second (Q) Brocard point

The most elegant construction of the Brocard points, described below using the example of the first Brocard point P (in the adjacent figure, the circles have been replaced by arcs for reasons of space), goes as follows:

One cuts the perpendicular bisector ms ₁ the side AB with the perpendicular s ₁ to the side BC through the point B . Draw a circle around the point of intersection so that it goes through point B. Then this circle also passes through the point A and touch the side BC at point B . Similarly, we construct a circle through points C and B that touches side CA at point C , and a circle through points A and C that touches side AB at point A. These three circles have a common point P - the first Brocard point of triangle ABC!

The three circles that have just been constructed are also referred to as additional circles of triangle ABC. The second Brocard point Q (green dashed lines) is constructed in the same way .

Formulas for the Brocard angle

If you write ABC for the area of the triangle, the Brocard angle can be calculated using the following formulas: ${\ displaystyle A _ {\ Delta}}$

${\ displaystyle \ tan \ omega = {\ frac {4A _ {\ Delta}} {a ^ {2} + b ^ {2} + c ^ {2}}}}$ .
${\ displaystyle \ cot \ omega = \ cot \ alpha + \ cot \ beta + \ cot \ gamma. \,}$
${\ displaystyle \ sin \ omega = {\ frac {2A _ {\ Delta}} {\ sqrt {b ^ {2} c ^ {2} + a ^ {2} c ^ {2} + a ^ {2} b ^ {2}}}}}$

For every triangle holds . ${\ displaystyle \ omega \ leq 30 ^ {\ circ}}$

properties

The two Brocard points of a triangle ABC are always isogonally conjugated to one another .
The center of the two Brocard points ( Kimberling number X (39)) lies on the so-called Brocard axis , which connects the circumcenter and the Lemoine point . The straight line connecting the two Brocard points is perpendicular to the Brocard axis.

Coordinates

First Brocard point
Trilinear coordinates	${\ displaystyle {\ frac {c} {b}} \,: \, {\ frac {a} {c}} \,: \, {\ frac {b} {a}}}$
Barycentric coordinates	${\ displaystyle {\ frac {ac} {b}} \,: \, {\ frac {ba} {c}} \,: \, {\ frac {cb} {a}}}$
Second Brocard point
Trilinear coordinates	${\ displaystyle {\ frac {b} {c}} \,: \, {\ frac {c} {a}} \,: \, {\ frac {a} {b}}}$
Barycentric coordinates	${\ displaystyle {\ frac {ab} {c}} \,: \, {\ frac {bc} {a}} \,: \, {\ frac {ca} {b}}}$

Third Brocard point

Occasionally the point is referred to as the "third" Brocard point. It has the Kimberling number and the barycentric coordinates , so it closes the circle with the first two Brocard points with the barycentric coordinates and . ${\ displaystyle \ alpha = a ^ {- 3}}$ ${\ displaystyle X_ {76}}$ ${\ displaystyle (a ^ {- 2}, b ^ {- 2}, c ^ {- 2})}$ ${\ displaystyle (b ^ {- 2}, c ^ {- 2}, a ^ {- 2})}$ ${\ displaystyle (c ^ {- 2}, a ^ {- 2}, b ^ {- 2})}$

literature

Ross Honsberger Episodes in Nineteenth and Twentieth Century Euclidean Geometry , MAA, 1995, Chapter 10 (Brocard Points)
Roger A. Johnson Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle . Boston, Houghton Mifflin 1929, reprinted as Advanced Euclidean Geometry , Dover 1960
Julian Coolidge A treatise on the geometry of the circle and the square , New York, Chelsea 1971

Web links

Eric W. Weisstein: Brocard Points ( First Brocard Point , Second Brocard Point , Third Brocard Point ) on MathWorld