Help with this hard geometry problem!

xxravenxx · #1 $Old$ October 20th, 2009, 12:38 AM

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.

Soroban · #2 $Old$ October 20th, 2009, 07:21 AM

Hello, xxravenxx!

Quote:

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.

xxravenxx · #3 $Old$ October 21st, 2009, 12:07 AM

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.

mr fantastic · #4 $Old$ October 21st, 2009, 05:35 AM

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.

xxravenxx · #5 $Old$ October 27th, 2009, 03:21 AM

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.