Given the semi-perimeter and the area of a Triangle, and also the volume of the cuboid whose edges are equal to the sides of the Triangle: find the sum of the squares of its sides.
If s = semi-perimeter, m = area, v = volume; then
Let s = semi-perimeter, m = area, v = volume.
We know that m = s · (s − a) · (s − b) · (s − c);
∴ m² = s · (s − a) · (s − b) · (s − c);
∴
m² |
s |
∴
m² |
s² |
v |
s |
∴ 2 ·(
m² |
s² |
v |
s |
∴ a² + b² + c² = 2 · (s² −
v |
s |
m² |
s² |