Pythagorean Triple - Some Relationships

Some Relationships

The radius, r, of the inscribed circle can be found by:

The unknown sides of a triple can be calculated directly from the radius of the incircle, r, and the value of a single known side, b.

k = b − 2r

a = 2r + (2 r2/k)

c = a+ k = 2r + (2r2 /k) + k

The solution to the "incircle problem" shows that, for any circle whose radius is a whole number r, setting k = 1, we are guaranteed at least one right-angled triangle containing this circle as its inscribed circle where the lengths of the sides of the triangle are a primitive Pythagorean triple:

a=2r + 2r2

b=2r + 1

c=2r + 2r2 + 1

The perimeter P and area K of the right triangle corresponding to a primitive Pythagorean triple triangle are

P = a + b + c = 2m(m + n)

K = ab/2 = mn(m2 − n2)

Additional relationships include:

where is the angle between the leg of length and the hypotenuse.

where is the angle between the leg of length 2mn and the hypotenuse.

If two numbers of a triple are known, the third can be found using the Pythagorean theorem.