Question

explain the heron's formula with proof

Answer 1

where b is the length of a base and h is the height to that base.




We have



so, for future reference, 

2s = a + b + c 
2(s - a) = - a + b + c 
2(s - b) = a - b + c 
2(s - c) = a + b - c

There is at least one side of our triangle for which the altitude lies "inside" the triangle. For convenience make that the side of length c. It will not make any difference, just simpler.





Let p + q = c as indicated. Then