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Math Help - Help with a geometry problem

  1. #1
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    Post Help with a geometry problem

    Given:
    BAL is a right triangle at A.
    O is the midpoint of [BL].
    The circle of diameter [BO] cuts [BA] at I.

    Show that I is the midpoint of [AB].
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  2. #2
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    Quote Originally Posted by Pythagorean Theorem View Post
    Given:
    BAL is a right triangle at A.
    O is the midpoint of [BL].
    The circle of diameter [BO] cuts [BA] at I.

    Show that I is the midpoint of [AB].
    1. Triangle BIO is a right triangle with \angle(BIO) = 90^\circ because I is placed on the Thales circle over BO.

    2. You have 2 similar triangles: \Delta(BAL) \sim \Delta(BIO)

    3. Use proportions.
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  3. #3
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    Here is a second way.
    The \angle OIB being inscribed in a semi-circle is a right angle.
    Therefore, \overleftrightarrow {AL}||\overleftrightarrow {BI} so by the midpoint theorem I is the midpoint of \overline {AB} .
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