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

• 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

• 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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