Perfect Triangles are defined as the
triangles having side lengths that are integers and for which the area and perimeter
of the triangle are equal.
Imagine a triangle with
side lengths of x + y, y + z, and x + z, the problem can be narrowed down
algebraically. For this triangle, the perimeter is equal to 2 (x +y+ z)
And the semi perimeter ‘s’ = (x +y+ z)
Equating area with the semi
perimeter we get:
√[s{s-(x +y)}{s-(y +z )} {s-( z+x)}]
= 2 (x +y+ z)
√ [(x +y+ z){(x +y+ z) -(x +y)}{(x +y+
z) -(y+z )}{ (x +y+ z) -(z+x)}]= 2(x +y+ z)
Now solving the above equation , we
get the following equation:
xyz = 4(x+y+z)
We have already solved the above
equation in the previous post on
Diophantine Equation:
The solutions to the above equation
are as below:
x
|
y
|
z
|
x+y
Side 1 of the Triangle
|
y+z
Side 2 of the Triangle
|
z+x
Side 3 of the Triangle
|
24
|
5
|
1
|
29
|
6
|
25
|
14
|
6
|
1
|
20
|
7
|
15
|
9
|
8
|
1
|
17
|
9
|
10
|
10
|
3
|
2
|
13
|
5
|
12
|
6
|
4
|
2
|
10
|
6
|
8
|
Thus there are only five Perfect
Triangles for which The Perimeter and the area are same numerically as given in the above table.