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

Centroid, Incenter and Circumcenter of a
Triangle using Vector Notation
Kurt Nalty
December 25, 2007
Abstract
This is a note providing formulas for the centroid, incenter and circumcenter
of a triangle, given three points, using vector notation.
Vector notation formulas for basic geometry are not common on the
web. I needed the formulas for the centroid, inscribed circle and circumscribed
circle given a triangle for a design project, but could not find the
inscribed and circumscribed formulas.
Having re-derived these formulas, I am posting them for convenience.
Point Definitions
Define three points by ⃗ P 1, ⃗ P 2, and ⃗ P 3. From these, define the directed line
segments and lengths:
⃗A = ⃗ P 2 − ⃗ P 1
⃗B = ⃗ P 3 − ⃗ P 2
⃗C = ⃗ P 1 − ⃗ P 3
Notice these are defined cyclically such that ⃗ A + ⃗ B + ⃗ C = 0.
Calculate lengths, semiperimeter and area
a = | ⃗ A|
b = | ⃗ B|
c = | ⃗ C|
s = (a + b + c)/2
K = √ s ∗ (s − a) ∗ (s − b) ∗ (s − c)
In vector form, the area is one half the cross product of any two sides.
⃗K = (1/2) ⃗ A × ⃗ B
= (1/2) ⃗ B × ⃗ C
= (1/2) ⃗ C × ⃗ A
1

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