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Olympiad workbookIOQMProperties of Triangles

Let ABC be a triangle and let be its circumcircle. The internal bisectors of angles A, B and C intersect at A ₁, ~B ₁ and C ₁ respectively and the internal bisectors of angles A ₁, ~B ₁ and C ₁ of the triangle A ₁ ~B ₁ C ₁ intersect at A ₂, ~B ₂ and C ₂ , respectively. If the smallest angle of triangle ABC is 40^ , what is the magnitude of the smallest angle of triangle A ₂ ~B ₂ C ₂ in degrees?

Correct answer

55

Step-by-step solution

aligned & A ₁ ~B ₁ C ₁= 2 - ABC 2 & ~A ₁ C ₁ ~B ₁= 2 - ACB 2 & ~B ₁ ~A ₁ C ₁= 2 - BAC 2 & ~A ₂ ~B ₂ C ₂= 2 - ( 2 - ABC 2 ) 2 & = 4 + ABC 4 aligned similarly A ₂ C ₂ ~B ₂= 4 + ACB 4 and B ₂ ~A ₂ C ₂= 4 + BAC 4 smaller angle of A₂ B₂ C₂ is 45^ + ( 40^ 4 )=55^

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