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The coordinates of the centroid relate to the average of coordinates of the triangle's vertices. Free online calculation tool - mathopenref.com/coordcentroid.html
a circle
If by sperical triangle you mean a triangle on the surface of a sphere, you will need 3 dimensional coordinate geometry. Whether you use polar coordinates or linear coordinates will depend on what you want to "solve".
That depends on where the triangle ABC is located on the Cartesian plane for the coordinates of its vertices to be determined.
Not too sure of the question but if A is (1, 2) and B is (-3, -1) then it is a right angle triangle if the coordinates of C are at (1, -1) or (-3, 2)