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 Chord-Product (Posted on 2018-01-27)

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 No Rating 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

 Subject Author Date Solution armando 2018-01-31 06:44:55

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