Friday, November 18, 2011

Problem 689: Triangle, Three Excircles, Tangency points, Tangent lines, Concurrent Lines, Mind Map

Geometry Problem
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to see the complete problem 689.

Online Geometry Problem 689: Triangle, Three Excircles, Tangency points, Tangent lines, Concurrent Lines.
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1 comment:

  1. PravinNovember 19, 2011 at 4:54 AM

    Let A₃A∩BC=X, B₃B∩AC=Y, C₃C∩AB=Z
    Y,A₁,C₂, lie respectively on the sides CA, AB, BC
    of ∆ABC such that YB, CA₁,AC₂ concur at B₃.
    ∴By Ceva’s Theorem,
    (AY/YC)(CC₂/C₂B)(BA₁/A₁A) = 1
    ∴ AY/YC = (C₂B/CC₂).(A₁A/BA₁)
    =[(s-a)/s].[s/(s-c)]
    = (s-a)/(s-c)
    Similarly,
    BZ/ZA = (s-b)/(s-a) and
    CX/XB = (s-c)/(s-b)
    ∴ (AY/YC).(BZ/ZA).(CX/XB)=1
    Hence by Converse of Ceva’s Theorem,
    AA₃(X), BB₃(Y), CC₃(Z) are concurrent

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