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Trigonometry with altitudes (Posted on 2023-07-09) Difficulty: 3 of 5
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).

No Solution Yet Submitted by Danish Ahmed Khan    
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re: Something's missing? | Comment 2 of 3 |
(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))

  Posted by Jer on 2023-07-09 17:09:53
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