Kept hearing the phrase "Bary-bash" which if you're from Australia might sound like a crime but if you are solving geometry problems it could in some situations be even more potent a tool than usual coordinate geometry methods (or Coord Bash). The Bary was for Barycentric coordinates, which started coming up from time to time and now we finally go through the background thoroughly. See my geometry course for where this fits in the grand scheme of things but in short we define coordinates using a triangle instead of x and y axes for example and this makes for solving problems which feature a prominent triangle very natural.

Barycentric coordinates is also the natural coordinate system for Projective geometry, which also uses homogeneous coordinates. As there are no circles, a reference triangle is ideal.

Simple problems from various contests are used as examples of how no problem is safe from being solved by Barycentric coordinates.

Example from IMO may also be available upon request but whats there should be enough for now.

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