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Math Help - the proof of law of sines

  1. #1
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    the proof of law of sines

    hi guys i need help on this question

    Prove that, for any triangle ABC inscribed in a circle of radius r,

    a over sinA= b over sinB= c over sinC= 2r.
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  2. #2
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    Quote Originally Posted by mathishard View Post
    hi guys i need help on this question

    Prove that, for any triangle ABC inscribed in a circle of radius r,

    a over sinA= b over sinB= c over sinC= 2r.
    Ah! The elegant extended law of sines.
    Below is a hint how to solve problem. I drew a line from a vertex through the diameter (which is not one of the vertices) until it intersected the circumfurence, in blue. THen I extended the red line. Note the two angles which are marked are equal because they create equal arcs.
    The rest I leave to thee.
    Attached Thumbnails Attached Thumbnails the proof of law of sines-picture3.gif  
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  3. #3
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    i know how to draw the triangle and stuff already it just that i don't know how to proof it.
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  4. #4
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    1)Since DC is the diameter we know that <DAC is right.
    2)Also, B=D.
    3)Thus, sin B= sin D
    4)But, sin D=AC/DC
    5)Thus, sin B=AC/DC
    6)Thus, 1/sin B=DC/AC
    7)Thus, AC/sin B=DC
    8)But, DC=2r
    9)Thus, AC/sin B=2r
    10)Similarly, AB/sin C=2r and BC/sin A=2r
    11)Thus, BC/sin A=AC/sin B=AB/sin C=2r
    ---
    This shows that side divided by opposite angle is 2r.
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