Reference Triangle

  • ΔABC
  • Vertices A, B, and C
  • Side lengths a, b, c
  • Interior angles α, β, γ
  • Semiperimeter s = (a+b+c)/2
  • Area K = sqrt(s⋅(s-a)⋅(s-b)⋅(s-c))

Inscribed Circle

  • Radius: r = K/s
  • Center: 1:1:1 (trilinear)
    O = (Ox,Oy) = ( (aAx+bBx+cCx)/(a+b+c), (aAy+bBy+cCy)/(a+b+c) )
  • The incenter is the intersection of the three lines of angle bisection
  • The incircle touches the triangle at three points
  • From vertices A, B, and C , the distances to the points of tangency are s-a, s-b, and s-c, respectively

Circumscribed Circle

  • Radius: R = (a⋅b⋅c/(4⋅K)
  • Center: cos(α):cos(β):cos(γ) (trilinear)
    O = (Ox,Oy) = ( see circumcenter )
  • The circumcenter is the intersection of the three lines of side bisection, i.e., perpendicular bisectors of the sides
  • The circumcircle touches the triangle at three points
    These points are the vertex points A, B, and C
  • From vertices A, B, and C the distances to the points of tangency (along the lines AB, BC, and CA) are c, a, and b, respectively

Exscribed Circles

  • There are three excircles for every triangle, denoted ΓA, ΓB, and ΓC
  • Radius: rA = K/(s-a), rB = K/(s-b), rC = K/(s-c)
  • Center: OA: -1:0:0, OB: 0:-1:0, OC: 0:0:-1, (trilinear)
  • Each excenter is the intersection of three angle bisectors. Excenter OA, for example, lies at the intersection of the interior angle bisector of A plus the two exterior angle bisectors of B, and C
  • The circumcircle touches the reference triangle at one point and the extensions of the other two sides at two other points.
  • The distances to the points of tangency are:
    • For OA:
      • From vertex A, along the line AB: s
      • From vertex B, along the line CB: s-c
      • From vertex C, along the line AB: s-b
    • For OB:
      • From vertex B, along the line BC: s
      • From vertex C, along the line CA: s-a
      • From vertex A, along the line AB: s-c
    • For OC:
      • From vertex C, along the line CA: s
      • From vertex A, along the line AB: s-b
      • From vertex B, along the line BC: s-a
  • K = sqrt(r⋅rB⋅rC⋅rA)
    • K is the area of the reference triangle
    • The areas of the extriangles are:
      • KA = (TBD)
      • KB = (TBD)
      • KC = (TBD)
  • 1/r = 1/rA + 1/rB + 1/rC

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