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D , E , F are three points on the sides ...

`D , E , F` are three points on the sides `B C ,C A ,A B ,` respectively, such that `/_A D B=/_B E C=/_C F A=thetadot` `A^(prime), B ' C '` are the points of intersections of the lines `A D ,B E ,C F` inside the triangle. Show that are of ` A^(prime)B^(prime)C^(prime)=4cos^2theta,` where `` is the area of ` A B Cdot`

Text Solution

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From `Delta AB'C (AB')/(sin(pi-(A + theta))) = (AC)/(sin(pi - B))`

`rArr AB' = 2R sin (A + theta)`
From `Delta AC'B, (AC')/(sin (theta - A)) = (AB)/(sin (pi - C))`
`rArr AC' = 2R sin (theta - A)`
`:. B'C' = 2R (sin (A + theta) - sin (theta - A))`
`= 4R cos theta sin A = 2a cos theta`
Similarly, `C'A' = 2b cos theta`
`:. " Area of " Delta A'B'C' = (1)/(2) (B'C') (A'C') sin angleB'C'A'`
`= (1)/(2) (2 a cos theta) (2b cos theta) sin C`
`= 4 cos^(2) theta xx Delta`
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