Hello sologuitar Originally Posted by
sologuitar Given: Isosceles triangle ABC with segment AB congruent to segment AC, and line CD is perpendicular to AB.
Prove: Segment BC multiplied by segment BC is equal to two times Segment AB multiplied by Segment BD.
Using Pythagoras' Theorem on $\displaystyle \triangle ADC$:
$\displaystyle DC^2=AC^2-AD^2$$\displaystyle = AB^2 -AD^2$, since $\displaystyle AC = AB$ (given)
$\displaystyle =(AB+AD)(AB-AD)$
$\displaystyle =(AB+AD)(BD)$ ... (1)
And using Pythagoras' Theorem on $\displaystyle \triangle BDC$:
$\displaystyle BC^2 = BD^2+DC^2$$\displaystyle = BD^2 + (AB+AD)(BD)$, from (1)
$\displaystyle =BD(BD +AB+AD)$
$\displaystyle =BD(AB+AB)$
$\displaystyle =2BD.AB$
Grandad