have gotten a bit closer now, i post my try in the hope someone can spot an error or finish the problem

i have joined C to O, centre of hemisphere so A0=0B=OC=a. this gives some isoc trinagle which lead to ACB=90,ABC=60.

resoloving vertically

Tsin 30+Tsin 60=w

T=2W/(rt(3)+1)

now

sin 60=AC/2a so AC=art(3) hence extension is art(3)-4a/3=a(21rt(3)-28)/21

cos 60=BC/2a so BC=a hence extension is a-4a/7=a(21-12)/21=9a/21

giving total extension a[21rt(3)-28+9]/21=a[rt(3)-1]

hence using

=

=

=

which is close but no cigar. How can i rid myself of the 29/21?