TrilinearsRelatedToOthers

1. Trilinears related to other systems of projective coordinates

Trilinears are the projective coordinates system of the plane, corresponding to the projective base {A,B,C,I}, I being the incenter of the triangle, whose coordinates are (1,1,1). They are discussed in Trilinears.html .
Trilinears, together with barycentrics, are used in the modern geometry of the triangle mainly to prove coincidences. Below we summarize the transition functions between some additional coordinate systems related to the trilinears.
Among them the transformation to the projection-ratios-coordinates, which locate the point P by the three ratios DB/DC, EC/EA, FA/FB.

One can locate P using the other ratios QB/QC, RC/RA, SA/SB, related to trilinears by:

Finally one can locate P also from the ratios PQ/AQ, PR/BR, PS/CS, related to trilinears by:

{a,b,c} denote side-lengths and {A,B,C} angle-measures. {h_a,h_b,h_c} denote the altitudes of the triangle.

2. Pedal and Cevian triangles of a point

The pedal and Cevian triangles of a point are two triangles created by simple recipes from a triangle ABC and a point D. The pedal (see Pedal.html ) has for vertices the projections of P on the sides.
The Cevian has for vertices the traces of lines {AP,BP,CP} respectively on the opposite sides.

The trilinears of the traces {Q,R,S} can be computed easily from the trilinears (x,y,z) of P and the results of the first section.

Analogously the trilinears of the projections {D,E,F} of P can be easily computed.

1. Trilinears related to other systems of projective coordinates

2. Pedal and Cevian triangles of a point

See Also