perplexus.info :: Geometry : Trigonometry with altitudes

Trigonometry with altitudes (Posted on 2023-07-09)

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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Playing around drawing this with Geometer's sketchpad seems not to give a unique answer.

The smallest product is 1/2 when the triangle is isosceles right with hypotenuse BC (O,D coincide as do A,E)

The largest is about 0.636 when the triangle is isosceles with angles about 64,64,52. Ratio of unequal sides about 1.15

Posted by Jer on 2023-07-09 16:20:47

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