Proposed Problem
Click the figure below to see the complete Routh's Theorem 3 about Triangle, Cevians, Ratio, Areas.
See also:
Routh's Theorem 3
Level: High School, SAT Prep, College geometry
Sunday, January 10, 2010
Routh's Theorem 3: Triangle, Cevians, Ratio, Areas
Subscribe to:
Post Comments (Atom)
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)
-----------------------------------------
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