# Isosceles Triangle ABC

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• Dec 22nd 2009, 08:19 AM
sologuitar
Isosceles 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, 08:55 AM
Grandad
Hello sologuitar
Quote:

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 $\triangle ADC$:
$DC^2=AC^2-AD^2$
$= AB^2 -AD^2$, since $AC = AB$ (given)

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

$=(AB+AD)(BD)$ ... (1)
And using Pythagoras' Theorem on $\triangle BDC$:
$BC^2 = BD^2+DC^2$
$= BD^2 + (AB+AD)(BD)$, from (1)

$=BD(BD +AB+AD)$

$=BD(AB+AB)$

$=2BD.AB$
Grandad