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Let D, E, F be points on the sides B C, C A, A B of a triangle A B C , respectively. Suppose A D, B E, C F are concurrent at P . If P F / P C=2 / 3, P E / P B=2 / 7 and P D / P A=m / n , where m, n are positive integers with gcd (m, n)=1 , find m+n .

Correct answer

45

Step-by-step solution

aligned & P F P C = 2 3 & P F C F = 2 5 = ₁+ ₃ ...(i) aligned Similarly 2 9 = ₂+ ₆ ...(ii) and m m+n = ₄+ ₅ ...(iii) Adding (i), (ii) and (iii); we get aligned & 2 5 + 2 9 + m m+n =1 & m m+n =1- 28 45 & m m+n = 17 45 & 28 m=17 n & m n = 17 28 & m+n=45 aligned

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