The Euler formula, connecting the number of faces F, edges E and vertices V of a convex polyhedron 
in 3D is widely known.  It is  F - E + V = 2.


In our case, we have a polygonal grid on a plane; in this case, the Euler formula is 

    F - S + V = 1,     (1)

where F is the number of faces (minimal polygons of the grid); S is the number of their sides and
V is the number of vertices.


We will calculate the numbers S and V in the formula (1), and then use it to determine F, our major unknown.



Part 1.  Calculating the number of vertices V


    Let call our triangle ABC, by the names of its vertices.


    We have 4 lines from vertex A to the opposite side "a".
    We have 4 lines from vertex B to the opposite side "b".
    These two families of lines have 4 x 4 = 16 intersection points.


    Next, we have 4 lines from vertex A to the opposite side "a",
     and  we have 4 lines from vertex C to the opposite side "c".
    These two families of lines produce other 4 x 4 = 16 intersection points, different from 16 points above.

    
    Finally, we have 4 lines from vertex B to the opposite side "b",
        and  we have 4 lines from vertex C to the opposite side "c".
    These two families of lines produce other 4 x 4 = 16 intersection points, different from 32 points above.


    So, we have 16 + 16 + 16 = 48 intersection points INSIDE the triangle ABC.
    To it, we should add 5 + 5 + 5 = 15 intersection points along the PERIMETER of the triangle ABC.

    In all, there are  48 + 15 = 63 intersection points:  V = 63.



Part 2.  Calculating the number of sides S

    
    From vertex A to the opposite side "a" we have 4 interior lines inside triangle ABC.

    Interesting fact is that each of these lines has 9 elementary interior segments.

        You may count and check this fact on your own.

    It is not an accidental fact: 9 elementary interior segments in each of these lines are created
    by 4 + 4 = 8 intersection points of this line with the family of 4 lines  " from B to b "  and
    with the family of 4 lines  " from C to c ".

    So, the four lines  " from A to a "  give us 4*9 = 36 elementary segments.



    Similarly, from vertex B to the opposite side "b" we have 4 interior lines inside triangle ABC.

    Again, interesting fact is that each of these lines has 9 elementary interior segments.

        You may count and check this fact on your own.

    The reason is similar to the above case.

    So, the four lines  " from B to b "  give us 4*9 = 36 another elementary segments.



    Finally and similarly, we have another 36 elementary segments from the family of lines " from C to c ".


    In all, we just counted  36 + 36 + 36 = 108 elementary segments INSIDE the triangle ABC.

    Add to it 5 + 5 + 5 = 15 elementary segments along the PERIMETER of triangle ABC.


    So, the total number of elementary segments is 108 + 15 = 123:  S = 123.



Part 3.  Applying the Euler formula


    From the formula (1),  we have


        F = 1 + S - V = 1 + 123 - 63 = 61.


ANSWER.  The number of faces inside the triangle ABC (non-overlapping regions) is 61.