Question-185111

Question Number 185111 by emmanuelson123 last updated on 17/Jan/23

Commented by Eulerian last updated on 17/Jan/23

what does d (O_{1}, O_{2}) means ?

Commented by JDamian last updated on 17/Jan/23

that means the "distance between O1 and O2"

Commented by emmanuelson123 last updated on 17/Jan/23

It means the distance between the centers of the circle

Answered by HeferH last updated on 17/Jan/23

Commented by HeferH last updated on 17/Jan/23

W h e n t w o c i r c l e s a r e i n t e r n a l l y t a n g e n t,

t h e d i s t a n c e o f t h e c e n t e r s i s R - r, i f w e

m o v e t h e i n n e r c i r c l e b y a b i t s u c h t h a t i s n o t

t a n g e n t a n y m o r e, t h e i r c e n t e r s a r e g o i n g

t o b e a b i t c l o s e r, t h e r e f o r e d (O_{1}, O_{2}) < R - r

Commented by mr W last updated on 20/Jan/23

t h e i n c i r c l e a n d t h e c i r c u m c i r c l e o f

a t r i a n g l e c a n n e v e r t o u c h e a c h o t h e r .

Commented by HeferH last updated on 20/Jan/23

?

Commented by mr W last updated on 20/Jan/23

Answered by Eulerian last updated on 17/Jan/23

Commented by Eulerian last updated on 17/Jan/23

Intuitively, we know that

θ < 45 °

∴ \tan θ < \tan 45 °

\frac{d (O_{1}, O_{2})}{R - r} < 1

d (O_{1}, O_{2}) < R - r

Answered by mr W last updated on 20/Jan/23

{e u l e r}^{'} s t h e o r e m :

d^{2} = R^{2} - 2 R r

d^{2} < R^{2} - 2 R r + r^{2} = {(R - r)}^{2}

\Rightarrow d < R - r

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