Trilinear Coordinates

I'm familiar with Cartesian coordinates and polar coordinates but only recently encountered trilinear coordinates in the context of triangles. See Figure 1.

Figure 1

In Figure 1, it can be seen that the point P is a perpendicular distance a', b' and c' from the sides a, b and c respectively. Working with Cartesian coordinates would require that the triangle be placed and oriented on the Cartesian plane. However, our interest is really on the distances a', b' and c' and even here it's not that actual distances but ratio of the distances. This is where trilinear coordinates come to our assistance.

The trilinear coordinates are written as ka' : kb' : kc' where k is any positive constant, chosen so that the ratio is in its simplest form. Point P in any triangle that is similar to that shown in Figure 1 can be associated with these trilinear coordinates. Figure 2 shows a different triangle in which the point P is external to the triangle.

Figure 2

Here the line AC or b lies between P and the vertex B. However, the lines AB or c and BC or a do not lie between the respective vertices C and A. In such a situation, kb' is given a negative sign whereas ka' and kc' are positive. Here are the trilinear coordinates for some common points:

• A = 1 : 0 : 0
  
  
• B = 0 : 1 : 1
  
  
• C = 0 : 0 : 1
  
  
• incentre = 1 : 1 : 1
  (the centre of the circle that has the triangle sides as tangents)
  
  
• centroid = bc : ca : ab = 1/a : 1/b : 1/c = cosecA : cosecB : cosecC
  (the point where the medians meet)
  
  
• circumcentre = cosA : cosB : cosC
  (the centre of the circle that passes through the three vertices)
  
  
• orthocentre = secA : secB : secC
  (the point where the altitudes meet)

There are many more points associated with triangles but this is a start. Note how the trilinear coordinates or trilinears as they are sometimes called can be expressed in terms of the sides of the triangle or the angles opposite these sides. Figure 3 shows the situation for the centroid (marked G) where AX, BY and CZ are medians.

Figure 3

Remember that the medians of a triangle, divide it into six smaller triangles of equal area. Looking at Figure 3, it can be seen that:

• Area of ∆AGC = Area of ∆AGB = Area of ∆BGC
• 1/2 x b x b' = 1/2 x c x c' = 1/2 x a x a'
• bb'=cc' and so b'/c'=c/b or b':c'=c:b or b':c'=ac:bc
• cc'=aa' and so c'/a'=a/c or c':a'=a:c or c':a'=ab:bc
• a':b':c'=bc:ac:ab=1/a:1/b:1/c
• a/sinA=b/sinB=c/sinC and so a:sinA=b:sinB=c:sinC
• a':b':c'=1/a:1/b:1/c=1/sinA:1/sinB:1/sinC=cosecA:cosecB:cosecC

Figure 4 shows a specific example of the trilinear coordinates for a right-angled triangle with angles of 30° and 60°.

Figure 4

References: 

• Triangle Convertor for Cartesian, Trilinear and Barycentric Coordinates
• Trilinear coordinates (Wikipedia)
• Trilinear Coordinates (MathWorld)