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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?
  

See The Solution Submitted by Bractals    
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Solution Comment 1 of 1
The diameter of the circumcircle passing through I determine the opposite points M N in the circumcircle. Let be O the  center of the circumcircle. Then:
PI * IQ = MI * IN (chords theorem) = (R+OI) * (R-OI) = R^2-OI^2

By Euler's theorem the distance OI between incenter and circumcenter is OI^2= R * (R-2r). 

So: 
PI * IQ = 2rR

Edited on January 31, 2018, 6:48 am
  Posted by armando on 2018-01-31 06:44:55

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