Median of a planar triangle divides it into two equal-area smaller triangles. Also, the centroid point can be used to divide planar triangle into three equal-area triangles. Whereas, in spherical triangles, such similar geometrical constructions of median/centroid does not yield smaller spherical triangles of equal-area.
Not sure about a spherical triangle but the following paper, which describes constructing an equal-area grid on a sphere, might be useful for your question:
Area-preserving projections from hexagonal and triangular domains to the sphere and applications to electron back-scatter diffraction pattern simulations
D Roşca and M De Graef
Another useful forum for questions such as this is http://math.stackexchange.com/. Be sure to follow the policies of the forum before you ask a question on Math StackExchange though.. you will also have to describe your failed attempts in solving this question!
For two spherical triangles I think there are three answers, each one obtained by drawing a spherical segment from a vertex to the opposite side. For example if the sides have angles A, B and C and we draw a segment from the AB vertex to side C then the segment of C adjacent to A should have angle (B+C-A)/2 and the one adjacent to B should angle (A+C-B)/2. This is all based upon the spherical area formula: area = R^2(A+B+C-pi). Don't have answer for three triangles yet but it seems like there should only be one configuration.
Equal-area division of planar triangle using geometrical construction of median, centroid is convenient and relatively straight forward. Whereas, equal-area division of spherical triangle into two or three smaller spherical triangles is a non-trivial problem. A tedious geometrical method for dividing into three parts has been posted in the math exchange site (check the attached link), but a geometrical solution for two-part division seems to be unknown. As suggested by some researchers, analytical solution is possible using spherical trignometry concepts. The pdf document containing the steps in derivation of formula/equation for equal-area division is attached herewith.
http://math.stackexchange.com/questions/1730002/how-to-divide-a-spherical-triangle-into-three-equal-area-spherical-triangles
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