Question

In the isosceles triangle ABC, AB = AC and BD is the altitude through B. Prove that
BD² - CD² = 2CD × AD.​

Answer 1

Answer:

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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