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Question:

In a triangle PQR, P is the largest angle and cosP = 1/3. Further the incircle of the triangle touches the sides PQ, QR and RP at N, L and M respectively, such that the lengths of PN, QL and RM are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are) 18, 16, 24, 22.

16

24

18

22

Solution:

Let s-a = 2k, s-b = 2k, s-c = 2k+2, k ∈ I, k>1
Adding we get, s = 6k
So a = 4k+2, b = 4k, c = 4k-2
Now, cosP = 1/3 ⇒ b² + c² - a² / 2bc = 1/3
⇒ 3[(4k)² + (4k-2)² - (4k+2)²] = 2 × 4k(4k-2)
⇒ 3[16k² + 16k² - 16k + 4 - (16k² + 16k + 4)] = 8k(4k-2)
⇒ 3[16k² - 32k] = 32k² - 16k
⇒ 48k² - 96k = 32k² - 16k
⇒ 16k² = 80k
⇒ k = 5
So, sides are 22, 20, 18