Anonymous
Anonymous asked in Education & ReferenceHomework Help · 7 years ago

Can Some on give me a proof of apollonius theorem?

Hi,

I saw a clear explanation with example for apollonius theorem in easycalculation.Where they have explained with step by step solution.

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  • 7 years ago
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    i) Apollonius Theorem:

    "In a triangle ABC, if AD is the median to BC, then AB² + AC² = 2(AD² + BD²)"

    ii) As I am not sure of your knowledge on Cosine Law of triangle, let me provide the solution using basic geometrical concepts applied in high school geometry.

    iii) Draw an altitude from vertex A to BC, to meet BC at E. [E may be within B & D or withing D & C, depends upon AB < AC or AB > AC; let us here take AB < AC, so E is between B & D]

    iv) Applying Pythagoras theorem to respective right triangles,

    From right triangle, AEB, AB² = AE² + BE² ------ (1)

    From right triangle, AED, AD² = AE² + ED² ------ (2)

    (1) - (2): AB² - AD² = BE² - ED² = (BD - DE)² - ED² = BD² - 2BD*DE + ED² - ED²

    ==> AB² - AD² = BD² - 2BD*DE ------- (3)

    v) Similarly considering the right triangles, AEC & AED, we get,

    AC² - AD² = DE² - ED² = (CD + DE)² - DE² = CD² + 2CD*DE + DE² - DE²

    ==> AC² - AD² = CD² + 2CD*DE = BD² + 2BD*DE ------ (4)

    [Since AD is the median, D is mid point of BC; hence BD = DC]

    vi) Adding (3) & (4)

    ==> AB² + AC² - 2AD² = 2BD²

    ==> AB² + AC² = 2(AD² + BD²) [Proved]

    • 6 years agoReport

      USE COSINE LAW OF TRIANGLE

  • 3 years ago

    Apollonius Theorem

  • 7 years ago

    I know two ways of proving. The first is based on cosine theorem. Choose any angle of triangle and express side and mediana using cosine theorem. Then equal cosines:

    m^2=AB^2+(AC/2)^2 - 2AB*AC/2*cosA

    BC^2=AB^2+AC^2 - 2*AB*AC*cosA

    2)Using the theorem about parallelogram. (a^2+b^2)=d1^2+d2^2, where d1 and d2 are diagonals of par.

    If you continue mediana until it's twice of its length and connect the end of mediana with two other vertexes you'll get a parallelogramm. One of its diagonals equals 2m.

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