**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

- From vertex
- 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

- From vertex
- 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

- From vertex

- For
- 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)

- K

- 1/r = 1/rA + 1/rB + 1/rC