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Question Number 3943 by Rasheed Soomro last updated on 25/Dec/15
What is the area  of  overlapping region  of three circles of radii r_1  , r_2  , r_3  with their  respective centres C_1  , C_2  and C_3  when  r_1 +r_2 > C_1 C_2   ,   r_2 +r_3 > C_2 C_3   and  r_3 +r_1 > C_3 C_1 .  Note that C_i C_j  is the distance between centres  C_(i ) and  C_j  .
$$\mathcal{W}{hat}\:{is}\:{the}\:{area}\:\:{of}\:\:{overlapping}\:{region} \\ $$$${of}\:{three}\:{circles}\:{of}\:{radii}\:\boldsymbol{\mathrm{r}}_{\mathrm{1}} \:,\:\boldsymbol{\mathrm{r}}_{\mathrm{2}} \:,\:\boldsymbol{\mathrm{r}}_{\mathrm{3}} \:{with}\:{their} \\ $$$${respective}\:{centres}\:\boldsymbol{\mathrm{C}}_{\mathrm{1}} \:,\:\boldsymbol{\mathrm{C}}_{\mathrm{2}} \:{and}\:\boldsymbol{\mathrm{C}}_{\mathrm{3}} \:{when} \\ $$$$\boldsymbol{\mathrm{r}}_{\mathrm{1}} +\boldsymbol{\mathrm{r}}_{\mathrm{2}} >\:\boldsymbol{\mathrm{C}}_{\mathrm{1}} \boldsymbol{\mathrm{C}}_{\mathrm{2}} \:\:,\:\:\:\boldsymbol{\mathrm{r}}_{\mathrm{2}} +\boldsymbol{\mathrm{r}}_{\mathrm{3}} >\:\boldsymbol{\mathrm{C}}_{\mathrm{2}} \boldsymbol{\mathrm{C}}_{\mathrm{3}} \:\:{and} \\ $$$$\boldsymbol{\mathrm{r}}_{\mathrm{3}} +\boldsymbol{\mathrm{r}}_{\mathrm{1}} >\:\boldsymbol{\mathrm{C}}_{\mathrm{3}} \boldsymbol{\mathrm{C}}_{\mathrm{1}} . \\ $$$${Note}\:{that}\:\boldsymbol{\mathrm{C}}_{\boldsymbol{\mathrm{i}}} \boldsymbol{\mathrm{C}}_{\boldsymbol{\mathrm{j}}} \:{is}\:{the}\:{distance}\:{between}\:{centres} \\ $$$$\boldsymbol{\mathrm{C}}_{\boldsymbol{\mathrm{i}}\:} {and}\:\:\boldsymbol{\mathrm{C}}_{\boldsymbol{\mathrm{j}}} \:. \\ $$
Commented by Yozzii last updated on 26/Dec/15
Commented by Yozzii last updated on 26/Dec/15
Yes. I agree. Look at my image  for a better idea on finding the full  area.
$${Yes}.\:{I}\:{agree}.\:{Look}\:{at}\:{my}\:{image} \\ $$$${for}\:{a}\:{better}\:{idea}\:{on}\:{finding}\:{the}\:{full} \\ $$$${area}. \\ $$
Commented by Rasheed Soomro last updated on 26/Dec/15
But the problem is to determine the central angles  of these arcs!
$${But}\:{the}\:{problem}\:{is}\:{to}\:{determine}\:{the}\:{central}\:{angles} \\ $$$${of}\:{these}\:{arcs}! \\ $$
Commented by Yozzii last updated on 25/Dec/15
It is possible that one circle completely  contains the other two overlapping  circles. The problem then requires  finding only the area in which the  two inner circles overlap since this  same area lies within the larger   circle. The conditions on r_i  and C_i C_j   are still satisfied.
$${It}\:{is}\:{possible}\:{that}\:{one}\:{circle}\:{completely} \\ $$$${contains}\:{the}\:{other}\:{two}\:{overlapping} \\ $$$${circles}.\:{The}\:{problem}\:{then}\:{requires} \\ $$$${finding}\:{only}\:{the}\:{area}\:{in}\:{which}\:{the} \\ $$$${two}\:{inner}\:{circles}\:{overlap}\:{since}\:{this} \\ $$$${same}\:{area}\:{lies}\:{within}\:{the}\:{larger}\: \\ $$$${circle}.\:{The}\:{conditions}\:{on}\:{r}_{{i}} \:{and}\:{C}_{{i}} {C}_{{j}} \\ $$$${are}\:{still}\:{satisfied}.\: \\ $$$$ \\ $$
Commented by Yozzii last updated on 25/Dec/15
I think I have a means of finding the required  area if no two circles fully contain each   other. It requires forming three new  circles and a triangle. A diagram   would truly help my explanation.     Suppose no two circles fully contain  each other and that they appear to form  eight distinct regions in the plane,  with the common region of overlap having three  vertices A,B,C. Construct the triangle  A,B,C. Call the area of △ABC A_1 .  We have AB,BC and CA as arcs,  because of the manner in which the  three cirvles overlap, and   we can form circles to adhere to  each of these arcs. Thus, if this is done  we can find the respective areas A_2 ,  A_3  and A_4 , that lie between each side  of the triangle and the arc opposite   each side. For a given side with an arc,  the area difference between the sector  of the circle with arc AB, for e.g,   and the isosceles triangle ABO_1 ,  where O_1  is the centre of this new   circle, is A_2 , for example. The area of  the region of overlap is then   A_1 +A_2 +A_3 +A_4 .
$${I}\:{think}\:{I}\:{have}\:{a}\:{means}\:{of}\:{finding}\:{the}\:{required} \\ $$$${area}\:{if}\:{no}\:{two}\:{circles}\:{fully}\:{contain}\:{each}\: \\ $$$${other}.\:{It}\:{requires}\:{forming}\:{three}\:{new} \\ $$$${circles}\:{and}\:{a}\:{triangle}.\:{A}\:{diagram}\: \\ $$$${would}\:{truly}\:{help}\:{my}\:{explanation}.\: \\ $$$$ \\ $$$${Suppose}\:{no}\:{two}\:{circles}\:{fully}\:{contain} \\ $$$${each}\:{other}\:{and}\:{that}\:{they}\:{appear}\:{to}\:{form} \\ $$$${eight}\:{distinct}\:{regions}\:{in}\:{the}\:{plane}, \\ $$$${with}\:{the}\:{common}\:{region}\:{of}\:{overlap}\:{having}\:{three} \\ $$$${vertices}\:{A},{B},{C}.\:{Construct}\:{the}\:{triangle} \\ $$$${A},{B},{C}.\:{Call}\:{the}\:{area}\:{of}\:\bigtriangleup{ABC}\:\boldsymbol{{A}}_{\mathrm{1}} . \\ $$$${We}\:{have}\:{AB},{BC}\:{and}\:{CA}\:{as}\:{arcs}, \\ $$$${because}\:{of}\:{the}\:{manner}\:{in}\:{which}\:{the} \\ $$$${three}\:{cirvles}\:{overlap},\:{and} \\ $$$$\:{we}\:{can}\:{form}\:{circles}\:{to}\:{adhere}\:{to} \\ $$$${each}\:{of}\:{these}\:{arcs}.\:{Thus},\:{if}\:{this}\:{is}\:{done} \\ $$$${we}\:{can}\:{find}\:{the}\:{respective}\:{areas}\:\boldsymbol{{A}}_{\mathrm{2}} , \\ $$$$\boldsymbol{{A}}_{\mathrm{3}} \:{and}\:\boldsymbol{{A}}_{\mathrm{4}} ,\:{that}\:{lie}\:{between}\:{each}\:{side} \\ $$$${of}\:{the}\:{triangle}\:{and}\:{the}\:{arc}\:{opposite}\: \\ $$$${each}\:{side}.\:{For}\:{a}\:{given}\:{side}\:{with}\:{an}\:{arc}, \\ $$$${the}\:{area}\:{difference}\:{between}\:{the}\:{sector} \\ $$$${of}\:{the}\:{circle}\:{with}\:{arc}\:{AB},\:{for}\:{e}.{g},\: \\ $$$${and}\:{the}\:{isosceles}\:{triangle}\:{ABO}_{\mathrm{1}} , \\ $$$${where}\:{O}_{\mathrm{1}} \:{is}\:{the}\:{centre}\:{of}\:{this}\:{new}\: \\ $$$${circle},\:{is}\:\boldsymbol{{A}}_{\mathrm{2}} ,\:{for}\:{example}.\:{The}\:{area}\:{of} \\ $$$${the}\:{region}\:{of}\:{overlap}\:{is}\:{then}\: \\ $$$$\boldsymbol{{A}}_{\mathrm{1}} +\boldsymbol{{A}}_{\mathrm{2}} +\boldsymbol{{A}}_{\mathrm{3}} +\boldsymbol{{A}}_{\mathrm{4}} .\: \\ $$$$ \\ $$
Commented by Rasheed Soomro last updated on 25/Dec/15
All possible cases should be considered  separately
$${All}\:{possible}\:{cases}\:{should}\:{be}\:{considered} \\ $$$${separately} \\ $$
Commented by Yozzii last updated on 25/Dec/15
I just realized that you only need the  three initial circles to do the  calculation. The new circles I speak of  only need to be the initial circles we   started with! (Waste of energy...)
$${I}\:{just}\:{realized}\:{that}\:{you}\:{only}\:{need}\:{the} \\ $$$${three}\:{initial}\:{circles}\:{to}\:{do}\:{the} \\ $$$${calculation}.\:{The}\:{new}\:{circles}\:{I}\:{speak}\:{of} \\ $$$${only}\:{need}\:{to}\:{be}\:{the}\:{initial}\:{circles}\:{we}\: \\ $$$${started}\:{with}!\:\left({Waste}\:{of}\:{energy}…\right) \\ $$
Commented by Rasheed Soomro last updated on 25/Dec/15
Three coplaner circles can form  atmost  seven distinct regions. I think.
$$\mathcal{T}{hree}\:{coplaner}\:{circles}\:{can}\:{form}\:\:{atmost} \\ $$$${seven}\:{distinct}\:{regions}.\:{I}\:{think}. \\ $$
Commented by Yozzii last updated on 25/Dec/15
The region outside all of them make it 8.
$${The}\:{region}\:{outside}\:{all}\:{of}\:{them}\:{make}\:{it}\:\mathrm{8}. \\ $$
Commented by Rasheed Soomro last updated on 26/Dec/15
You  should start with 7−region   case , I think.  I also  think that the area of overlapping region is closely/direcly  related to the quantities:   r_1 +r_2 −C_1 C_2  , r_2 +r_3 −C_2 C_3  , r_3 +r_1 −C_3 C_1   More the circles are closed greater the area   of overlapping region.
$${You}\:\:{should}\:{start}\:{with}\:\mathrm{7}−{region}\: \\ $$$${case}\:,\:{I}\:{think}. \\ $$$${I}\:{also}\:\:{think}\:{that}\:{the}\:{area}\:{of}\:{overlapping}\:{region}\:{is}\:{closely}/{direcly} \\ $$$${related}\:{to}\:{the}\:{quantities}:\: \\ $$$$\boldsymbol{\mathrm{r}}_{\mathrm{1}} +\boldsymbol{\mathrm{r}}_{\mathrm{2}} −\boldsymbol{\mathrm{C}}_{\mathrm{1}} \boldsymbol{\mathrm{C}}_{\mathrm{2}} \:,\:\boldsymbol{\mathrm{r}}_{\mathrm{2}} +\boldsymbol{\mathrm{r}}_{\mathrm{3}} −\boldsymbol{\mathrm{C}}_{\mathrm{2}} \boldsymbol{\mathrm{C}}_{\mathrm{3}} \:,\:\boldsymbol{\mathrm{r}}_{\mathrm{3}} +\boldsymbol{\mathrm{r}}_{\mathrm{1}} −\boldsymbol{\mathrm{C}}_{\mathrm{3}} \boldsymbol{\mathrm{C}}_{\mathrm{1}} \\ $$$${More}\:{the}\:{circles}\:{are}\:{closed}\:{greater}\:{the}\:{area}\: \\ $$$${of}\:{overlapping}\:{region}. \\ $$
Commented by Yozzii last updated on 26/Dec/15
A,C &E are circle centres. To find  the area between the arc HI and the  chord HI, find the area of △CHI  and then the area of the sector  CHI. The positive difference between  the two areas gives what we′re   searching for. This is similarly  done for sector EGH and sector AGI.
$${A},{C}\:\&{E}\:{are}\:{circle}\:{centres}.\:{To}\:{find} \\ $$$${the}\:{area}\:{between}\:{the}\:{arc}\:{HI}\:{and}\:{the} \\ $$$${chord}\:{HI},\:{find}\:{the}\:{area}\:{of}\:\bigtriangleup{CHI} \\ $$$${and}\:{then}\:{the}\:{area}\:{of}\:{the}\:{sector} \\ $$$${CHI}.\:{The}\:{positive}\:{difference}\:{between} \\ $$$${the}\:{two}\:{areas}\:{gives}\:{what}\:{we}'{re}\: \\ $$$${searching}\:{for}.\:{This}\:{is}\:{similarly} \\ $$$${done}\:{for}\:{sector}\:{EGH}\:{and}\:{sector}\:{AGI}. \\ $$$$ \\ $$

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