Question

Fie triunghiul ABC dreptunghic in A si AD inaltime .Aratati ca:
$\frac{1}{ AD^{2}} = \frac{1}{ AB^{2} } + \frac{1}{ AC^{2} }$

Answer 1

$\frac{1}{AD ^{2} } = \frac{1}{AB ^{2} } + \frac{1}{AC ^{2} }$ ⇔
$\frac{1}{AD ^{2} } = \frac{AC ^{2} }{AC ^{2}*AB ^{2} } + \frac{AB ^{2} }{AB ^{2} *AC ^{2} }$⇔
$\frac{1}{AD ^{2} } = \frac{AC ^{2} + AB ^{2} }{AC ^{2}*AB ^{2} }$⇔
$\frac{1}{AD ^{2} } = \frac{BC ^{2} }{ 4*AriaABC }$⇔

4*AriaABC = BC² * AD² ⇔

4*AriaABC = 4*AriaABC - > Adevarat

Deci $\frac{1}{AD ^{2} } = \frac{1}{AB ^{2} } + \frac{1}{AC ^{2} }$