ABC is an acute triangle with circumcenter O and altitudes AD and BE.
If O lies on segment DE, find the value of sin(A)sin(B)sin(C).
(In reply to
Something's missing? by Jer)
That maximum value is (sqrt(5)1)sqrt(2+sqrt(5))/4 = 0.6360098248
Found by coordinate geometry. A=(0,1), B=(0,1), C=(c,0)
xcoordinate of O=(c^21)/(2c)
xcoordinate of D=(2c)/(c^2+1)
Solving gives c=sqrt(2+sqrt(5))
sin(A)=sin(B)=sqrt(2+sqrt(5))/sqrt(3+sqrt(5))
sin(C)=2sqrt(2+sqrt(5))/(3+sqrt(5))

Posted by Jer
on 20230709 17:09:53 