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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- Oct 19th 2009, 10:38 PMxxravenxxHelp 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. - Oct 20th 2009, 05:21 AMSoroban
Hello, xxravenxx!

Quote:

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`

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A o * \

| θ * *\ E

| θ o

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| * \ θ ' *

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O * * \ *

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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.

- Oct 20th 2009, 10:07 PMxxravenxx
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. (Rofl) - Oct 21st 2009, 03:35 AMmr fantastic
- Oct 27th 2009, 01:21 AMxxravenxx