Geometry Problem

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

Click the figure below to see the complete problem 689.

## Friday, November 18, 2011

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

Labels:
concurrent,
excircle,
Gergonne,
Nagel theorem,
tangent,
triangle

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Let A₃A∩BC=X, B₃B∩AC=Y, C₃C∩AB=Z

ReplyDeleteY,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