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.

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- Dec 22nd 2009, 07:19 AMsologuitarIsosceles Triangle ABC
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.

- Dec 22nd 2009, 07:55 AMGrandad
Hello sologuitarUsing Pythagoras' Theorem on $\displaystyle \triangle ADC$:

$\displaystyle DC^2=AC^2-AD^2$And using Pythagoras' Theorem on $\displaystyle \triangle BDC$:$\displaystyle = AB^2 -AD^2$, since $\displaystyle AC = AB$ (given)

$\displaystyle =(AB+AD)(AB-AD)$

$\displaystyle =(AB+AD)(BD)$ ... (1)

$\displaystyle BC^2 = BD^2+DC^2$Grandad$\displaystyle = BD^2 + (AB+AD)(BD)$, from (1)

$\displaystyle =BD(BD +AB+AD)$

$\displaystyle =BD(AB+AB)$

$\displaystyle =2BD.AB$