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Sixty is Special (Posted on 2017-08-18) Difficulty: 4 of 5
I is the incenter of triangle ABC with B' the intersection of the ray BI and side AC and C' the intersection of ray CI and side AB. Prove that AB'IC' is a cyclic quadrilateral if and only if angle BAC is 60 degrees.

  Submitted by Bractals    
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Solution: (Hide)

I first tried to prove that AB'IC' was cyclic if and only if the
point I lies on the circumcircle of ΔAB'C', but that started to
generate too much algebra.
The following is more geometrical and the difficulty level
drops from 4 to 2.

Let α, β, and γ be the measures of ∠BAC, ∠CBA, and ∠ACB
respectively. ∠AB'I and ∠AC'I are exterior angles of ΔB'CB
and ΔC'BC respectively. Thus,

   ∠AB'I = ∠B'CB + ∠CBB' = γ + β/2   and
   ∠AC'I = ∠C'BC + ∠BCC' = β + γ/2   and


   α = 60°  <==>  β + γ = 120°
                 <==>  3*(β + γ)/2 = 180°
                 <==>  (γ + β/2) + (β + γ/2) = 180°
                 <==>  ∠AB'I + ∠AC'I = 180°
                 <==>  AB'IC' is cyclic.


Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionSolution & a further resultJer2017-08-27 17:58:50
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