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In A B C , let L ,M ,N be the feet of t...

In ` A B C ,` let `L ,M ,N` be the feet of the altitudes. The prove that `sin(/_M L N)+sin(/_L M N)+sin(/_M N L)=4sinAsinBsinC`

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Using properties of pedal triangle, we have
`angle MLN = 180^(@) - 2A`
`angle LMN = 180^(@) - 2B`
`anlge MNL = 180^(@) - 2C`
`rArr sin (angle MLN) + sin (angle LMN) + sin (angle MNL)`
`= sin 2A + sin 2B + sin 2C`
`= 4 sin A sin B sin C`
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