The Sutherland-Hodgman clipping algorithm is used here to compute the intersection of two Delaunay triangulations with one another. Sutherland-Hodgman implementation by Joy Patel.
This algorithm can be used to intersect the decomposition of one triangle with a decomposition coming from another triangle of equal area.
For a decomposition of two triangles of equal area that uses Sutherland-Hodgman clipping, see this example.
See d3-equidecompose.