Prove that any altitude of a triangle must be less than the semiperimeter.
If the altitude h from A divides a into two parts, x and y, then since any side of a triangle is less than the sum of the other two sides,
h < b+x
h < c+y
2h < b+c+x+y
but, x+y = a, so
2h < b+c+a
h < (a+b+c)/2