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)
x-coordinate of O=(c^2-1)/(2c)
x-coordinate 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))
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Posted by Jer
on 2023-07-09 17:09:53 |
re: Something's missing?
