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23 July, 16:00

The line segment s bisects the angle C so that the two angles labeled x are equal. Show that the length of this angle bisector is s = 2abcosx/a + b

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  1. 23 July, 17:10
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    If we have a triangle with sides a and b and an included angle of C, then the area of the triangle would be:

    A = (1/2) ab sin C

    If angle C is bisected into two each angles each measuring x, then the area can be expressed as:

    A = (1/2) ab sin 2x

    Using the trigonometric identity for sin 2x = 2 sin x cos x, the area would now be:

    A = ab sin x cos x

    Since the line segment s divides the angle into two, it also divides the triangle into two. Another equation for the area is:

    A = (1/2) as sin x + (1/2) bs sin x

    Equating the two equations gives us:

    ab cos x = (1/2) as + (1/2) bs

    Solving for s

    s = 2 ab cos x / (a + b)
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