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### Deriving Trilinear Coordinates

```
Date: 05/18/99 at 14:59:16
From: Alejandra Smith
Subject: Trilinear coordinates

How do you derive the trilinear coordinates of, for example, the
orthocenter of a triangle? (cos B, cos C)
```

```
Date: 05/18/99 at 15:44:33
From: Doctor Floor
Subject: Re: Trilinear coordinates

Hi, Alejandra,

Trilinear coordinates are coordinates w.r.t. a triangle:

Let P be a point in the plane of a triangle ABC. Drop perpendicular
altitudes from P to the triangle sides, to find A', B', and C':

C
/ \
/   \
B'    A'
/   P   \
/         \
A-----C'----B

Trilinear coordinates (trilinears) are now the distances (PA', PB',
PC'). In fact, the coordinates are not necessarily the distances
themselves, but the ratios of the distances. These are called
"homogeneous," which means that if you take coordinates (x,y,z) and
(p*x,p*y,p*z) for some real number p, then these two sets of
coordinates represent the same point.

You can read a short introduction on trilinear coordinates on
University of Evansville Professor Clark Kimberling's webpages, from
which you can find a lot of triangle centers listed:

http://cedar.evansville.edu/~ck6/tcenters/trilin.html

To find trilinears when P is the orthocenter, note that:

angle BAP = angle BAA' = 90 degrees - angle B
(angle BA'A is right)
angle CAP = 90 degrees - angle C

So:

cos(B) = sin(90-B) = sin(BAP) = PC'/AP
cos(C) = sin(90-C) = sin(CAP) = PB'/AP

We can conclude that:

1        1
PB':PC' = cos(C) : cos(B) = ------ : ------
cos(B)   cos(C)

In the same way we will find that:

1        1
PA':PB' = cos(B) : cos(A) = ------ : ------
cos(A)   cos(B)

This yields trilinear coordinates that are slightly different than the
ones you suggested:

1        1        1
( ------ , ------ , ------ ), or (sec(A), sec(B), sec(C)).
cos(A)   cos(B)   cos(C)

As an alternative for trilinear coordinates, you can read in the Dr.
Math archives on the related topic of "Barycentric calculus":

http://mathforum.org/dr.math/problems/noble1.6.99.html

I hope this helps!

Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Coordinate Plane Geometry
High School Geometry
High School Triangles and Other Polygons
High School Trigonometry

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