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Math Help - isosceles triangle

  1. #1
    Member disclaimer's Avatar
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    isosceles triangle

    Hi;

    The coordinates of the vertices of an isosceles triangle are integers. Show that the square of the length of its base is an even number.

    Any hint would be helphul. Thanks in advance.
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    Hello,
    Quote Originally Posted by disclaimer View Post
    Hi;

    The coordinates of the vertices of an isosceles triangle are integers. Show that the square of the length of its base is an even number.

    Any hint would be helphul. Thanks in advance.
    Let (x_1,y_1) be the coordinates of the summit : A_1
    Let (x_2,y_2) and (x_3,y_3) be the coordinates of the two other points : A_2,A_3

    We know that A_1A_2=A_1A_3. Hence A_1A_2^2=A_1A_3^2

    So this can be written :

    \boxed{(x_2-x_1)^2+(y_2-y_1)^2=(x_3-x_1)^2+(y_3-y_1)^2}


    Now you want to prove that A_2A_3^2 is even.

    A_2A_3^2=(x_3-x_2)^2+(y_3-y_2)^2

    Now write x_3-x_2=(x_3-x_1)+(x_1-x_2) and y_3-y_2=(y_3-y_1)+(y_1-y_2)

    and expand the two squares following this model : \left((x_3-x_1)+(x_1-x_2)\right)^2=(x_3-x_1)^2+(x_1-x_2)^2+2(x_1-x_2)(x_3-x_1)


    Then simplify and conclude
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