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Question Number 152647 by mr W last updated on 30/Aug/21

Commented by mr W last updated on 30/Aug/21

$f i n d \frac{F G}{G B} = ?$
	Answered by mr W last updated on 30/Aug/21

	Commented by mr W last updated on 30/Aug/21

	$l e t A B = a, B C = b$ $H A = H C = H E = R$ $G E = G D = G B = r$ $\frac{A B}{B F} = \frac{B F}{B C}$ $\Rightarrow {B F}^{2} = A B \times B C = a b . . . (i)$ $R = \frac{A C}{2} = \frac{a + b}{2}$ $H B = a - R = \frac{a - b}{2}$ ${H G}^{2} = {H E}^{2} - {G E}^{2} = R^{2} - r^{2} = {(\frac{a + b}{2})}^{2} - r^{2}$ ${H G}^{2} = {H B}^{2} + {B G}^{2} = {(\frac{a - b}{2})}^{2} + r^{2}$ $\Rightarrow {(\frac{a + b}{2})}^{2} - r^{2} = {(\frac{a - b}{2})}^{2} + r^{2}$ $\Rightarrow {(\frac{a + b}{2})}^{2} - {(\frac{a - b}{2})}^{2} = 2 r^{2}$ $\Rightarrow 2 r^{2} = a b . . . (i i)$ ${B F}^{2} = 2 r^{2}$ $\Rightarrow B F = \sqrt{2} r = \sqrt{2} G B$ $\Rightarrow F G + G B = \sqrt{2} G B$ $\Rightarrow \frac{F G}{G B} = \sqrt{2} - 1$
	Commented by Tawa11 last updated on 30/Aug/21

	$Weldone sir . Thanks . God bless you more .$
	Commented by Tawa11 last updated on 30/Aug/21

	$Am learning from all your geometry solution .$ $I just start from beginning .$
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