Question

IN a triangle ABC, let H, I and O be the orthocentre, incentre and circumcentre, respectively. If the points B, H, I, C lie on a circle, how to get magnitude of ∠OBC in degrees?

Answer 1

$\huge\star\underline\mathfrak\red{Answer:-}$

In △ABC , O is orthocentre

By property of circle,∠BOC=2∠A      From fig.1

In fig.2

In □APHQ

∠A+∠APH+∠PHQ+∠HQA=360∘

∠HQA=∠APH=90∘

⟹∠A+∠PHQ=180∘

∠PHQ=∠BHC  (Vertically opposite angle)

⟹∠A+∠BHC=180∘

⟹∠A=−∠BHC+180∘

In fig.3

∠A+∠B+∠C=180∘

∠AIB=x=2∠C

Similarly, ∠BIC=y=2∠B

So, ∠A+2x+2y=180∘

⟹2∠A=90−(x+y)

In △CIB

∠BIC+x+y=180∘

⟹2∠A=90

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