Let ABC be a triangle with sides 51,52,53 . Let denote the incircle of ABC . Draw tangents to which are parallel to the sides of ABC . Let r ₁, r ₂, r ₃ be the inradii of the three corner triangles so formed. Find the largest integer that does not exceed r₁+r₂+r₃ .

Step-by-step solution

Let PQ be one of the required tangents is parallel to BC and meets sides AB and AC at P and Q respectively. Further Let P Q=x and B C=51 Now ABC is similar to APQ so x a = r ₁ r aligned Similarly & y 6 = r₂ r Hence & r₁+r₂+r₃ r = x a + y b + z c Now & x a = h-2 r h where h is altitude aligned array ll & x a =1- 2 s 2 , x=1- a s & x a =3- a+b+c s =1 so & r₁+r₂+r₃=r Now & r= 78(27)(26)(25) 78 & = 3^3 5^2 2.13 2 3 13 & = 3^2 5^2 =15 array