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 Existing: Non-Existing: