Question Number 104468 by ajfour last updated on 21/Jul/20

Commented byajfour last updated on 21/Jul/20

Find r and hence area of △ABC in  terms of R.

Commented bymr W last updated on 22/Jul/20

it′s not clear how the small circle is  unquely defined. the blue circle in  the diagram is also valid?

Commented bymr W last updated on 22/Jul/20

Commented bymalwaan last updated on 22/Jul/20

mr W  note that  ∣AB∣=∣AF ∣; ∣BF ∣=∣BE∣  in △BDE : ∡BED=90°  ;∣ED∣= r ; ∣BD∣=2R  ⇒∣BE∣=(√(4R^2 −r^2 ))  ....

Commented bymr W last updated on 22/Jul/20

then r is not unique, i think.

Commented byajfour last updated on 22/Jul/20

lets then have R and r given,  Sir; to find area of △ABC,  in terms of R and r.

Commented bymalwaan last updated on 22/Jul/20

it depend on the location  of the point A  SO you are absolutely right  mr W

Answered by mr W last updated on 22/Jul/20

Commented bymr W last updated on 22/Jul/20

sin β=(r/(2R))  ⇒β=sin^(−1) (r/(2R))  ⇒α=2β  AB=(R/(tan (α/2)))=((R(√(4R^2 −r^2 )))/r)  β+∠C=(π/2)−α  ∠C=(π/2)−α−β=(π/2)−3β  ((BC)/(sin α))=((AB)/(sin ∠C))=((AB)/(cos 3β))  ⇒BC=((sin 2β AB)/(cos 3β))  Δ_(ABC) =((AB×BC×sin ((π/2)+β))/2)  =((AB^2 ×sin 2β×cos β)/(2 cos 3β))  =((R^2 (4R^2 −r^2 ) sin β×cos^2  β)/(r^2  cos β (4 cos^2  β−3)))  =((R^2 (4R^2 −r^2 ) (r/(2R))(√(1−(r^2 /(4R^2 )))))/(r^2  (4(1−(r^2 /(4R^2 )))−3)))  =((R^2 (4R^2 −r^2 )^(3/2) )/(4r(R^2 −r^2 )))

Commented byajfour last updated on 24/Jul/20

thanks sir, shall review it, bit  unwell again..  Elegant Solving Sir, too good to  follow it even, thanks a lot!

Commented bymr W last updated on 23/Jul/20

get well!