The Three Kinds of Circles Associated with a Triangle (thing)

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 Γ_A, Γ_B, and Γ_C
• Radius: rA = K/(s-a), rB = K/(s-b), rC = K/(s-c)
• Center: O_A: -1:0:0, O_B: 0:-1:0, O_C: 0:0:-1, (trilinear)
  
• Each excenter is the intersection of three angle bisectors. Excenter O_A, 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 O_A:
    • 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 O_B:
    • 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 O_C:
    • 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:
    • K_A = (TBD)
    • K_B = (TBD)
    • K_C = (TBD)
• 1/r = 1/rA + 1/rB + 1/rC