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Olympiad workbookIOQMProperties of Triangles

Let A B C be a triangle with B A C=90^ and D be the point on the side B C such that A D B C . Let r, r₁ , and r₂ be the inradii of triangles A B C, A B D and A C D , respectively. If r, r₁ , and r₂ are positive integers and one of them is 5 , find the largest possible value of r+r₁+r₂ .

Correct answer

30

Step-by-step solution

A B D A C D A B C And hypotenuse of these triangles follow the Pythagoras theorem. So r₁^2+r₂^2=r^2 If r₁, r₂, r N and least is 5 , then the other two must be 12 and 13 . (r+r₁+r₂ )_ =30

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