Barycentric coordinates
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Barycentric coordinates are triples of numbers corresponding to masses placed at the vertices of a reference triangle
. These masses then determine a point
, which is the geometric centroid of the three masses and is identified with coordinates
. The vertices of the triangle are given by
,
, and
. Barycentric coordinates were discovered by Möbius in 1827 (Coxeter 1969, p. 217; Fauvel et al. 1993).
The Central NC Math Group published a lecture concerning this topic at https://www.youtube.com/watch?v=KQim7-wrwL0 if you would like to view it.
Useful formulas
Notation
Let the triangle be a given triangle,
be the lengths of
We use the following Conway symbols:
is semiperimeter,
is twice the area of
where
is the inradius,
is the circumradius,
is the cosine of the Brocard angle,
Main
For any point in the plane there are barycentric coordinates(BC):
:
The normalized (absolute) barycentric coordinates NBC satisfy the condition
they are uniquely determined:
Triangle vertices
The barycentric coordinates of a point do not change under an affine transformation.
Lines
The straight line in barycentric coordinates (BC) is given by the equation
The lines given in the BC by the equations and
intersect at the point
These lines are parallel iff
The sideline contains the points
its equation is
The line has equation
it intersects the sideline
at the point
Iff then
Let NBC of points and
be
Then the square of distance
The equation of bisector of
is:
Nagel line :
Circles
Any circle is given by an equation of the form
Circumcircle contains the points
the equation of this circle:
The incircle contains the tangent points of the incircle with the sides:
The equation of the incircle is
where
The radical axis of two circles given by equations of this form is:
Conjugate
The point is isotomically conjugate with respect to
with the point
The point is isogonally conjugate with respect to
with the point
The point is isocircular conjugate with respect to
with the point
Triangle centers
The median centroid is
The simmedian point is isogonally conjugate with respect to
with the point
The bisector the incenter is
The excenters are
The circumcenter lies at the intersection of the bisectors
and
its BC coordinates
The orthocenter is isogonally conjugate with respect to
with the point
Let Nagel point lies at line
The Gergonne point is the isotomic conjugate of the Nagel point, so
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Product of isogonal segments
Let triangle the circumcircle
and isogonals
and
of the
be given.
Let point
and
be the isogonal conjugate of a point
and
with respect to
Prove that
Proof
We fixed and the point
So isogonal
is fixed.
Denote
We need to prove that do not depends from
Line has the equation
To find the point we solve the system this equation and equation for circumcircle:
We use the formula for isogonal cobnjugate point and get
and then
We calculate distances (using NBC) and get:
where
has sufficiently big formula.
Therefore
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