Fermat Point - Properties

Properties

• When the largest angle of the triangle is not larger than 120°, X(13) is the Fermat point.
• The angles subtended by the sides of the triangle at X(13) are all equal to 120° (Case 2), or 60°, 60°, 120° (Case 1).
• The circumcircles of the three regular triangles in the construction intersect at X(13).
• Trilinear coordinates for the first isogonic center, X(13):

csc(A + π/3) : csc(B + π/3) : csc(C + π/3), or, equivalently,

sec(A − π/6) : sec(B − π/6) : sec(C − π/6).

• Trilinear coordinates for the second isogonic center, X(14):

csc(A − π/3) : csc(B − π/3) : csc(C − π/3), or, equivalently,

sec(A + π/6) : sec(B + π/6) : sec(C + π/6).

• Trilinear coordinates for the Fermat point:

1 − u + uvw sec(A − π/6) : 1 − v + uvw sec(B − π/6) : 1 − w + uvw sec(C − π/6)

where u, v, w respectively denote the Boolean variables (A<120°), (B<120°), (C<120°).

• The isogonal conjugate of X(13) is the first isodynamic point, X(15):

sin(A + π/3) : sin(B + π/3) : sin(C + π/3).

• The isogonal conjugate of X(14) is the second isodynamic point, X(16):

sin(A − π/3) : sin(B − π/3) : sin(C − π/3).

• The following triangles are equilateral:

antipedal triangle of X(13)

antipedal triangle of X(14)

pedal triangle of X(15)

pedal triangle of X(16)

circumcevian triangle of X(15)

circumcevian triangle of X(16)

• The lines X(13)X(15) and X(14)X(16) are parallel to the Euler line. The three lines meet at the Euler infinity point, X(30).
• The points X(13), X(14), circumcenter, nine-point center lie on a Lester circle.