The sides and height of a triangle are 4 consecutive integers.
Evaluate the triangle's area.
I will assume the altitude is the smallest of the four consecutive numbers. Let the altitude be x, then the sides are x+1, x+2, and x+3.
The area can be expressed by Heron's Formula:
If s=(a+b+c)/2 then the area is sqrt[s*(s-a)*(s-b)*(s-c)]
Applying this to our triangle yields A=(1/4)*sqrt[(3x+6)*(x)*(x+2)*(x+4)]. The area can also be expressed as A=x*y/2 where y is one of x+1, x+2, or x+3. Then:
(1/4)*sqrt[(3x+6)*(x)*(x+2)*(x+4)] = x*y/2.
Squaring each side and simplifying:
4x^2*y^2 = 3*x*(x+2)^2*(x+4)
Let y=x+2. Then the equation simplifies to 4x^2 = 3x^2 + 12x, which has a positive integer solution of x=12. One solution triangle has sides 13,14,15 and altitude 12.
Substituting y=x+1 and y=x+3 did not lead to integer solutions.