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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?
$$\mathrm{what}\:\mathrm{does}\:{d}\left({O}_{\mathrm{1}} ,\:{O}_{\mathrm{2}} \right)\:\mathrm{means}? \\ $$
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
When two circles are internally tangent,    the distance of the centers is R − r, if we    move the inner circle by a bit such that is not   tangent anymore, their centers are going   to be a bit closer, therefore d(O_1 , O_2 ) < R− r
$${When}\:{two}\:{circles}\:{are}\:{internally}\:{tangent},\: \\ $$$$\:{the}\:{distance}\:{of}\:{the}\:{centers}\:{is}\:{R}\:−\:{r},\:{if}\:{we}\: \\ $$$$\:{move}\:{the}\:{inner}\:{circle}\:{by}\:{a}\:{bit}\:{such}\:{that}\:{is}\:{not} \\ $$$$\:{tangent}\:{anymore},\:{their}\:{centers}\:{are}\:{going} \\ $$$$\:{to}\:{be}\:{a}\:{bit}\:{closer},\:{therefore}\:{d}\left({O}_{\mathrm{1}} ,\:{O}_{\mathrm{2}} \right)\:<\:{R}−\:{r} \\ $$$$\: \\ $$
Commented by mr W last updated on 20/Jan/23
the incircle and the circumcircle of  a triangle can never touch each other.
$${the}\:{incircle}\:{and}\:{the}\:{circumcircle}\:{of} \\ $$$${a}\:{triangle}\:{can}\:{never}\:{touch}\:{each}\:{other}. \\ $$
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°         ((d(O_1 , O_2 ))/(R−r)) < 1        d(O_1 , O_2 ) < R−r      ■
$$\:\mathrm{Intuitively},\:\mathrm{we}\:\mathrm{know}\:\mathrm{that} \\ $$$$\:\theta\:<\:\mathrm{45}° \\ $$$$\:\therefore\:\:\:\mathrm{tan}\:\theta\:<\:\mathrm{tan}\:\mathrm{45}° \\ $$$$\:\:\:\:\:\:\:\frac{{d}\left({O}_{\mathrm{1}} ,\:{O}_{\mathrm{2}} \right)}{{R}−{r}}\:<\:\mathrm{1} \\ $$$$\:\:\:\:\:\:{d}\left({O}_{\mathrm{1}} ,\:{O}_{\mathrm{2}} \right)\:<\:{R}−{r}\:\:\:\:\:\:\blacksquare \\ $$
Answered by mr W last updated on 20/Jan/23
euler′s theorem:  d^2 =R^2 −2Rr  d^2 <R^2 −2Rr+r^2 =(R−r)^2   ⇒d<R−r
$${euler}'{s}\:{theorem}: \\ $$$${d}^{\mathrm{2}} ={R}^{\mathrm{2}} −\mathrm{2}{Rr} \\ $$$${d}^{\mathrm{2}} <{R}^{\mathrm{2}} −\mathrm{2}{Rr}+{r}^{\mathrm{2}} =\left({R}−{r}\right)^{\mathrm{2}} \\ $$$$\Rightarrow{d}<{R}−{r} \\ $$

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