Proposed Problem

Click the figure below to see the complete Routh's Theorem 3 about Triangle, Cevians, Ratio, Areas.

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Routh's Theorem 3

Level: High School, SAT Prep, College geometry

## Sunday, January 10, 2010

### Routh's Theorem 3: Triangle, Cevians, Ratio, Areas

Labels:
area,
Ceva's theorem,
cevian,
Menelaus' theorem,
ratio,
Routh's theorem,
triangle

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SABA'/S = 1/1+k ( see P1) (1)

ReplyDeletedraw from B h1 altitude on C'A', and h2 from A on C'A'

=>h1/h2 = n

SAA'C' = C'A'∙h2 = C'A'∙h1/n

SBC'A' = C'A'∙h1

=>

SAA'C'/SBC'A' = 1/n

=>

SABA'/SBC'A' = (1+n)/n

=>

(1/(1+k))/SBC'A' = (1+n)/n ( SABA' = 1/1+k from (1)

SBC'A' = n/(1+k)(1+n)

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S1=Area(A’BC’) = ½ *BC’*BA’*sin(B) =1/2 *n*f*d*sin(B)

ReplyDeleteS=Area(ABC)=1/2*AB*BC*sin(B)=1/2*(1+n)*f*(1+k)*d*sin(B)

So S1/S=n/[(n+1)*(k+1)]

Calculate the same way for other ratios

Peter Tran