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Math Help - Help with this hard geometry problem!

  1. #1
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    Help with this hard geometry problem!

    Problem 14
    Let ABC be a right angled triangle. A circle Г have side AC as its diameter meets hypotenuse AB at point E. A tangent line to Г at point E meets side BC at point D. Prove that a triangle BDE is isosceles.
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  2. #2
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    Hello, xxravenxx!

    14. Let ABC be a right triangle: . \angle C = 90^o.
    A circle O has side AC as its diameter, meets hypotenuse AB at E.
    A tangent line to O at E meets side BC at point D.

    Prove that \Delta BDE is isosceles.
    Code:
              F
               o
                \
               θ'\
          A o *   \
            | θ * *\ E
            |     θ o
            |     *  *  *
            |   *     \ θ ' *
            | *       *\        *
          O *         * \           *
            |         *  \              *
            |             \                 *
            |        *     \                    *
            |       *       \                       *
            |     *          \                       θ' *
          C o * - - - - - - - o - - - - - - - - - - - - - - o B
                                D

    Draw radius OE.
    Since OA = OE,\;\Delta AOE is isosceles.
    . . \angle OAE = \angle OEA = \theta

    Since D{E}F is tangent at E,\;\angle OEF = 90^o.
    . . Hence, \angle OEA and \angle AEF are complementary.
    . . Let \angle AEF = \theta'

    \angle AEF and \angle DEB are vertical angles: . {\color{blue}\angle DEB = \theta'}


    In \Delta ABC,\;\angle CAB= \theta and \angle ABC are complementary.
    . . Hence: . {\color{blue}\angle ABC = \theta'}


    In \Delta BDE\!:\;\angle DEB = \angle EBD = \theta'

    Therefore, \Delta BDE is isosceles.

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  3. #3
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    Awesome. Thanks heaps. But a little advice - try to draw a diagram through paint or something instead of typing it up. It makes it a little hard to understand. But I got there.

    Thank you.
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  4. #4
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    Quote Originally Posted by xxravenxx View Post
    Awesome. Thanks heaps. But a little advice - try to draw a diagram through paint or something instead of typing it up. It makes it a little hard to understand. But I got there.

    Thank you.
    *Ahem* It's better than nothing ..... Be thankful (and I see you were ) for small favours.
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  5. #5
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    Quote Originally Posted by mr fantastic View Post
    *Ahem* It's better than nothing ..... Be thankful (and I see you were ) for small favours.

    No no, I'm just saying. It would be easier for you to draw up a pic in paint then type up a picture.
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