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Math Help - Proof: A Semicircle and Three Tangents

  1. #1
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    Proof: A Semicircle and Three Tangents


    Hypothesis: AB is the diameter, the line a passes through A and is tangent to the semicircle, the line b passes through B and is tangent to the semicircle, the line EF passes through P and is tangent to the semicircle. P is any point of the semicircle and E and F are the intersections of the line passing through P and tangent to the semicircle with line a and b.
    Thesis: PE * PF = (1/4) * AB^2 .

    I hope I explained well the problem, otherwise I'll write it again! Thank you!
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  2. #2
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    Re: Proof: A Semicircle and Three Tangents

    Quote Originally Posted by goby View Post

    Hypothesis: AB is the diameter, the line a passes through A and is tangent to the semicircle, the line b passes through B and is tangent to the semicircle, the line EF passes through P and is tangent to the semicircle. P is any point of the semicircle and E and F are the intersections of the line passing through P and tangent to the semicircle with line a and b.
    Thesis: PE * PF = (1/4) * AB^2 .

    I hope I explained well the problem, otherwise I'll write it again! Thank you!
    Let O be the centre of the semicircle.

    Join OP, OE and OF.

    Using congruent triangles, it is easy to prove that \angle EOA=\angle EOP and \angle FOB=\angle FOP.

    So, \angle EOF=\angle EOP+\angle FOP=\frac{1}{2}(\angle AOP+\angle BOP)=\frac{1}{2}*180^o=90^o.

    Note that OP\perp EF.

    So, PE*PF=OP^2 (This is a well-known theorem that can be proved using similar triangles.)

    PE*PF=\left(\frac{1}{2}AB\right)^2=\frac{1}{4}AB^2

    QED
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    Re: Proof: A Semicircle and Three Tangents

    It wasn't so easy! However I've understood it now, thank you. You meant Euclid's Second Theorem, din't you?
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    MHF Contributor alexmahone's Avatar
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    Re: Proof: A Semicircle and Three Tangents

    Quote Originally Posted by goby View Post
    It wasn't so easy! However I've understood it now, thank you.
    You're welcome!

    You meant Euclid's Second Theorem, din't you?
    No. (Euclid's second theorem states that the number of primes is infinite. - Wolfram MathWorld)
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