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Chord-Product (Posted on 2018-01-27) Difficulty: 3 of 5

  
Let Γ be the circumcircle of ΔABC with radius R and let I and r
be its incenter and inradius respectively.

If PQ is an arbitrary chord of Γ that passes through I, then what is
the value of the product |PI|*|IQ| in terms of R and r?
  

  Submitted by Bractals    
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Solution: (Hide)

  
For convenience let ∠A = 2α, ∠B = 2β, and ∠C = 2γ. Construct
chords AA', BB', and CC' passing through I and thus bisecting
angles A, B, and C respectively. Construct line segment FI ⊥ AB
where point F lies on AB. Construct diameter A'E and chords EB
and BA'.

   ∠CBA' = ∠CAA' = α
   ∠IBA' = ∠IBC + ∠CBA' = β + α
   ∠BIA' = ∠IAB + ∠IBA = α + β
      ∴ ∠IBA' = ∠BIA';
      ∴ |BA'| = |IA'|

   ΔAFI ∼ ΔEBA'
      since ∠AFI = 90° = ∠EBA' and ∠IAF = α = ∠A'EB

   ∴ |AI|/r = |AI|/|FI| = |EA'|/|BA'| = 2R/|IA'|
        or
      |AI|*|IA'| = 2*R*r

   and ∴ |PI|*|IQ| = 2*R*r by the Chord Theorem

QED
  

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Solutionarmando2018-01-31 06:44:55
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