Tri, Tri Again, Again

Place a collection of right triangles into the 9 x 9 grid below.

• Every triangle has integer-length legs, and both legs lie along grid lines (so one leg is horizontal and one is vertical, and all three corners are grid points). The hypotenuse may cut diagonally through cells.
• Every triangle must lie entirely within the 9 x 9 grid.
• Each triangle must contain exactly one number, and that number is the triangle's area. Every number must be contained in exactly one triangle.
• The entire 1 x 1 cell holding the number must lie inside its triangle (the cell may touch the triangle's boundary).
• Triangles' interiors may not overlap (their edges and corners may touch). Consequently, a triangle's interior may not cover any part, however small, of another number's cell, since that cell lies wholly inside the other number's triangle.
• Parts of the grid may be left uncovered by any triangle.

(Rows r0 to r8 run top to bottom, columns c0 to c8 run left to right.)

What to submit: the product of the odd horizontal leg lengths in the completed grid, taking one factor per triangle (so if two triangles both have horizontal leg 7, the factor 7 appears twice). Legs of even length, and all vertical legs, are ignored. Give a single integer.