Given any triangle $\triangle ABC$, let us draw its orthocenter $D$. By means of this point, we can draw three circles with centers in $A,B,C$ and passing through $D$.
These circles intersect in the points $E,F,G$, which can be seen as the vertices of three triangles $\triangle AFC, \triangle CGB$ and $\triangle BEA$.
My conjecture is that
The sum of the areas of the triangles $\triangle AFC, \triangle CGB, \triangle BEA$ is equal to the area of the triangle $\triangle ABC$.
If we substitute the orthocenter $D$ with the centroid of $\triangle ABC$, then the areas of $\triangle AFC, \triangle CGB$ and $\triangle BEA$ are all equal, and their sum is equal to the area of the triangle $\triangle ABC$.
Maybe these are very well known theorems. However, is there a compact proof for such conjectures?
NOTE: These conjectures are very similar to the one exposed in this post.
Thanks for your help, and sorry for imprecision or triviality.