Question

1.In an isosceles triangle ABC with AB=AC,D and E
are points on BC such that BE=CD.Show that AD=AE.
2.In ∆ABC,the bisector AD of ∠A is perpendicular to the side BC.Show
that AB=AC and ∆ABC is isosceles.

Answer 1

Using Pythagoras theorem:

   AD² + BD² = AC²   (= AB²)       --- (1)
  CD² + BD² = BC²                 ------- (2)

(2) -  2 * (1)  =>
=>  CD² - BD² =  BC² - 2 AC² + 2 AD²
=>                   =  BC² - 2 (AD +CD)² + 2 AD² 
=>                   =  BC² - 2 CD² - 4 AD * CD
=>                   =  BD² + CD²  - 2 CD² -4 AD * CD
=>                   = BD² - CD² - 4 AD * CD
Simplify to get 
     BD² - CD² = 2 AD * CD

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