Centroid, Incenter and Circumcenter of a Triangle using ... - Kurt Nalty

Centroid
The centroid is the true average coordinate. This center is the intersection of
the segment bisectors to opposite vertice.
⃗G = (1/3) ∗ ( ⃗ P 1 + ⃗ P 2 + ⃗ P 3) (1)
Incenter
The inscribed circle center is given by the weighted average of the coordinates
by the opposite side. The center is the intersection of the three angle bisectors.
The radius of the inscribed circle is
⃗I = (b ∗ ⃗ P 1 + c ∗ ⃗ P 2 + a ∗ ⃗ P 3)
a + b + c
(2)
r = √ (s − a) ∗ (s − b) ∗ (s − c)/s
= K/s
Circumcenter or Exocenter
The circumcenter or exocenter is found by erect perpendicular bisectors from
each side. This center is the intersection of these lengths. Building off the ⃗ P 1 -
⃗P 3 line, we have
⃗E = (1/2)( ⃗ P 1 + ⃗ P 3) + ⃗ A · ⃗B
8K 2 ( ⃗ C × ( ⃗ A × ⃗ B)) (3)
The radius of the circumscribed circle is
Notes
R = abc
4K
The equation of a line through a point is
⃗L = ⃗ P + s⃗u
where ⃗ P is the known point, s is the distance travelled on the line from that
point, and ⃗u is the direction of the line (as a unit vector).
The distance between a point ⃗ C and a line defined by two points ⃗ P 1 and
⃗P 2, as found by triangular area, is
d = |( ⃗ P 2 − ⃗ P 1) × ( ⃗ C − ⃗ P 1)|
| ⃗ P 2 − ⃗ P 1|
2