perplexus.info :: Geometry : Primitive Pythagorean Triangles

Primitive Pythagorean Triangles (Posted on 2008-01-26)

A primitive Pythagorean triangle (PPT) is a right triangle whose side lengths are integers that are relatively prime.

1) Prove that the inradius of a PPT has a different parity than the mean of the hypotenuse and the odd leg.

2) Prove that there exists an infinite number of pairs of non-congruent PPTs such that both members of the pair have the same inradius.

Submitted by Bractals

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Solution: (Hide)

A triangle ABC with a² + b² = c² is a PPT if and only if there exists a pair of integers (m,n) that are relatively prime with different parity such that

c = m² + n²
b = m² - n²
a = 2mn

Let r and s be the inradius and semiperimeter respectively of the PPT. Then
(m² + n²) + (m² - n²) + (2mn) s = ------------------------------ = m(m+n) 2
The area of the PPT is
ab (2mn)(m² - n²) rs = rm(m + n) = ---- = ---------------- 2 2
Therefore, r = n(m-n).

Part I:

Inradius is even <==> n(m - n) is even
<==> n is even
<==> m is odd
<==> m² is odd
<==> [ (m² + n²) + (m² - n²) ] / 2 is odd
<==> Mean of the hypoteneuse and the odd side is odd.

Part II:

let k > 1 be an integer. Then the pairs (2k,1) and (2k,2k-1) generate non-congruent PPTs with the same inradius 2k - 1.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re(2): solution	Daniel	2008-01-27 01:54:02
re: solution	Bractals	2008-01-26 17:00:04
solution	Daniel	2008-01-26 16:31:54

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