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| Chord-Product (Posted on 2018-01-27) |
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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?
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Submitted by Bractals
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Solution:
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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
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