Thread: Help with this hard geometry problem!

1. 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.

2. Hello, xxravenxx!

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

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

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

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

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

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

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

Therefore, $\displaystyle \Delta BDE$ is isosceles.

3. 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.

4. Originally Posted by xxravenxx
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.

5. Originally Posted by mr fantastic
*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.