A Beautiful Idea!!!!
by AlastorMoody, Jan 6, 2020, 3:15 PM
Today, I read some AoPS user named: invert dog about O. Let's try and see.
So here's our DOG
![[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(16cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -8.212743653345392, xmax = 7.193153269986815, ymin = -3.8608141981035144, ymax = 4.8902403657237; /* image dimensions */
pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451);
/* draw figures */
draw(circle((-5.94,2.31), 1.8965231345807523), linewidth(2) + wrwrwr); draw(shift((0.31581249353859664,0.7359161350801323))*rotate(176.43889953584588)*xscale(4.778969756221019)*yscale(1.5504337555473655)*unitcircle, linewidth(2) + wrwrwr); draw(shift((-3.204075361141203,-1.6940388925687504))*rotate(93.66135405716425)*xscale(1.606254467524427)*yscale(0.7977441148518869)*unitcircle, linewidth(2) + wrwrwr); draw(shift((3.3346282520460893,-2.0223251544029246))*rotate(-93.78969151511754)*xscale(1.4310473213896726)*yscale(0.5640549052639585)*unitcircle, linewidth(2) + wrwrwr); draw(shift((-1.0075277467308572,-2.1625554353210568))*rotate(102.78648964954485)*xscale(1.5578073923691285)*yscale(0.7503101192542945)*unitcircle, linewidth(2) + wrwrwr); draw(shift((4.580242174429869,-1.6304716489216273))*rotate(-90.41667113369833)*xscale(1.4608549628083582)*yscale(0.2401192721797382)*unitcircle, linewidth(2) + wrwrwr);
draw((-4.161103300856004,-3.2678286542183232)--(-3.1384733021773745,-3.2976544192694828), linewidth(2) + wrwrwr);
draw((-1.561135843475043,-3.662253751388024)--(-0.7579060713679635,-3.69054017564399), linewidth(2) + wrwrwr);
draw((2.380914886121239,-3.458631483841274)--(3.2129991384681165,-3.446781600828671), linewidth(2) + wrwrwr);
draw((4.0237281387010855,-3.3424455063602996)--(4.587800018484778,-3.087240118334038), linewidth(2) + wrwrwr);
/* dots and labels */
dot((-2.610799442261206,1.078371279450788),dotstyle);
label("$O$", (-2.5554962585479912,1.2029273316146858), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](//latex.artofproblemsolving.com/2/5/1/25171450ed8c757ca40efcb182666203faf80a8c.png)
Let's Invert,
![[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import contour; import graph; size(19cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -3.5377819143437375, xmax = -1.2568811508871698, ymin = 0.48763423406205636, ymax = 1.783260651336978; /* image dimensions */
pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451);
/* draw figures */
draw(circle((-2.8716806784776088,1.1748835855146615), 0.1486144470066057), linewidth(2) + wrwrwr);
real implicitf1 (real x, real y) { return 27802.497038878017-14641.22283749406*y^1+8497.77948897523*y^2-1766.3627782588546*y^3+393.78566976482205*y^4+35583.763204585906*x^1-9482.423964451282*x^1*y^1+4225.683746512857*x^1*y^2+17966.748544943664*x^2-1766.3627782588546*x^2*y^1+787.5713395296441*x^2*y^2+4225.683746512857*x^3+393.78566976482205*x^4; }
draw(contour(implicitf1, (xmin,ymin), (xmax,ymax), new real[]{0}, 500), linewidth(2) + wrwrwr);
real implicitf2 (real x, real y) { return -344.94426912701806+151.4616217417289*y^1-96.65634827571932*y^2+17.806976488294566*y^3-4.753790720756751*y^4-446.28480085815954*x^1+98.7038815365296*x^1*y^1-52.35710477872232*x^1*y^2-225.12569937941316*x^2+17.806976488294566*x^2*y^1-9.507581441513501*x^2*y^2-52.35710477872232*x^3-4.753790720756751*x^4; }
draw(contour(implicitf2, (xmin,ymin), (xmax,ymax), new real[]{0}, 500), linewidth(2) + wrwrwr);
real implicitf3 (real x, real y) { return -362.10876108684033+198.95902633343374*y^1-127.87200702004212*y^2+27.838646561003536*y^3-6.908883631946855*y^4-499.8000675089226*x^1+137.98240664562405*x^1*y^1-68.90413732732895*x^1*y^2-272.17549408478845*x^2+27.838646561003536*x^2*y^1-13.81776726389371*x^2*y^2-68.90413732732895*x^3-6.908883631946855*x^4; }
draw(contour(implicitf3, (xmin,ymin), (xmax,ymax), new real[]{0}, 500), linewidth(2) + wrwrwr);
real implicitf4 (real x, real y) { return -4025.1689414953134+2247.466401116625*y^1-1363.31779007272*y^2+292.99004018713254*y^3-68.40354945629488*y^4-5362.5336949219945*x^1+1496.9968208937714*x^1*y^1-699.0610392874046*x^1*y^2-2835.523355795609*x^2+292.99004018713254*x^2*y^1-136.80709891258977*x^2*y^2-699.0610392874046*x^3-68.40354945629488*x^4; }
draw(contour(implicitf4, (xmin,ymin), (xmax,ymax), new real[]{0}, 500), linewidth(2) + wrwrwr);
draw(shift((-2.613142385284668,0.9980392639307372))*xscale(0.08036617509577573)*yscale(0.08036617509577573)*arc((0,0),1,232.4076302018455,257.9192112110657), linewidth(2) + wrwrwr);
draw(shift((-2.613440607206124,1.0033719689120568))*xscale(0.07504580157178223)*yscale(0.07504580157178223)*arc((0,0),1,296.9867562023971,314.48275888114046), linewidth(2) + wrwrwr);
draw(shift((-2.6097092102082557,1.0018165217625874))*xscale(0.07656252040410111)*yscale(0.07656252040410111)*arc((0,0),1,4.6482724472355805,13.48892752260185), linewidth(2) + wrwrwr);
draw(shift((-2.5892964855903644,1.0308440180397496))*xscale(0.052165292319906444)*yscale(0.052165292319906444)*arc((0,0),1,-1.6974259224121713,5.543047336484977), linewidth(2) + wrwrwr);
real implicitf5 (real x, real y) { return -5467.5244403743445+3051.5483510183285*y^1-1841.795530656712*y^2+395.14672450848167*y^3-91.67876382775592*y^4-7255.536320223792*x^1+2024.7179009537413*x^1*y^1-939.5257246444035*x^1*y^2-3823.0611290972947*x^2+395.14672450848167*x^2*y^1-183.35752765551183*x^2*y^2-939.5257246444035*x^3-91.67876382775592*x^4; }
draw(contour(implicitf5, (xmin,ymin), (xmax,ymax), new real[]{0}, 500), linewidth(2) + wrwrwr);
/* dots and labels */
dot((-2.610799442261206,1.078371279450788),dotstyle);
label("$O$", (-2.6029865194844883,1.0971208315102876), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](//latex.artofproblemsolving.com/0/9/2/092060cb56af740942800cfcf037f9dbaf7de3c0.png)
This seems something really stupid
So here's our DOG
![[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(16cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -8.212743653345392, xmax = 7.193153269986815, ymin = -3.8608141981035144, ymax = 4.8902403657237; /* image dimensions */
pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451);
/* draw figures */
draw(circle((-5.94,2.31), 1.8965231345807523), linewidth(2) + wrwrwr); draw(shift((0.31581249353859664,0.7359161350801323))*rotate(176.43889953584588)*xscale(4.778969756221019)*yscale(1.5504337555473655)*unitcircle, linewidth(2) + wrwrwr); draw(shift((-3.204075361141203,-1.6940388925687504))*rotate(93.66135405716425)*xscale(1.606254467524427)*yscale(0.7977441148518869)*unitcircle, linewidth(2) + wrwrwr); draw(shift((3.3346282520460893,-2.0223251544029246))*rotate(-93.78969151511754)*xscale(1.4310473213896726)*yscale(0.5640549052639585)*unitcircle, linewidth(2) + wrwrwr); draw(shift((-1.0075277467308572,-2.1625554353210568))*rotate(102.78648964954485)*xscale(1.5578073923691285)*yscale(0.7503101192542945)*unitcircle, linewidth(2) + wrwrwr); draw(shift((4.580242174429869,-1.6304716489216273))*rotate(-90.41667113369833)*xscale(1.4608549628083582)*yscale(0.2401192721797382)*unitcircle, linewidth(2) + wrwrwr);
draw((-4.161103300856004,-3.2678286542183232)--(-3.1384733021773745,-3.2976544192694828), linewidth(2) + wrwrwr);
draw((-1.561135843475043,-3.662253751388024)--(-0.7579060713679635,-3.69054017564399), linewidth(2) + wrwrwr);
draw((2.380914886121239,-3.458631483841274)--(3.2129991384681165,-3.446781600828671), linewidth(2) + wrwrwr);
draw((4.0237281387010855,-3.3424455063602996)--(4.587800018484778,-3.087240118334038), linewidth(2) + wrwrwr);
/* dots and labels */
dot((-2.610799442261206,1.078371279450788),dotstyle);
label("$O$", (-2.5554962585479912,1.2029273316146858), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/2/5/1/25171450ed8c757ca40efcb182666203faf80a8c.png)
Let's Invert,
![[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import contour; import graph; size(19cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -3.5377819143437375, xmax = -1.2568811508871698, ymin = 0.48763423406205636, ymax = 1.783260651336978; /* image dimensions */
pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451);
/* draw figures */
draw(circle((-2.8716806784776088,1.1748835855146615), 0.1486144470066057), linewidth(2) + wrwrwr);
real implicitf1 (real x, real y) { return 27802.497038878017-14641.22283749406*y^1+8497.77948897523*y^2-1766.3627782588546*y^3+393.78566976482205*y^4+35583.763204585906*x^1-9482.423964451282*x^1*y^1+4225.683746512857*x^1*y^2+17966.748544943664*x^2-1766.3627782588546*x^2*y^1+787.5713395296441*x^2*y^2+4225.683746512857*x^3+393.78566976482205*x^4; }
draw(contour(implicitf1, (xmin,ymin), (xmax,ymax), new real[]{0}, 500), linewidth(2) + wrwrwr);
real implicitf2 (real x, real y) { return -344.94426912701806+151.4616217417289*y^1-96.65634827571932*y^2+17.806976488294566*y^3-4.753790720756751*y^4-446.28480085815954*x^1+98.7038815365296*x^1*y^1-52.35710477872232*x^1*y^2-225.12569937941316*x^2+17.806976488294566*x^2*y^1-9.507581441513501*x^2*y^2-52.35710477872232*x^3-4.753790720756751*x^4; }
draw(contour(implicitf2, (xmin,ymin), (xmax,ymax), new real[]{0}, 500), linewidth(2) + wrwrwr);
real implicitf3 (real x, real y) { return -362.10876108684033+198.95902633343374*y^1-127.87200702004212*y^2+27.838646561003536*y^3-6.908883631946855*y^4-499.8000675089226*x^1+137.98240664562405*x^1*y^1-68.90413732732895*x^1*y^2-272.17549408478845*x^2+27.838646561003536*x^2*y^1-13.81776726389371*x^2*y^2-68.90413732732895*x^3-6.908883631946855*x^4; }
draw(contour(implicitf3, (xmin,ymin), (xmax,ymax), new real[]{0}, 500), linewidth(2) + wrwrwr);
real implicitf4 (real x, real y) { return -4025.1689414953134+2247.466401116625*y^1-1363.31779007272*y^2+292.99004018713254*y^3-68.40354945629488*y^4-5362.5336949219945*x^1+1496.9968208937714*x^1*y^1-699.0610392874046*x^1*y^2-2835.523355795609*x^2+292.99004018713254*x^2*y^1-136.80709891258977*x^2*y^2-699.0610392874046*x^3-68.40354945629488*x^4; }
draw(contour(implicitf4, (xmin,ymin), (xmax,ymax), new real[]{0}, 500), linewidth(2) + wrwrwr);
draw(shift((-2.613142385284668,0.9980392639307372))*xscale(0.08036617509577573)*yscale(0.08036617509577573)*arc((0,0),1,232.4076302018455,257.9192112110657), linewidth(2) + wrwrwr);
draw(shift((-2.613440607206124,1.0033719689120568))*xscale(0.07504580157178223)*yscale(0.07504580157178223)*arc((0,0),1,296.9867562023971,314.48275888114046), linewidth(2) + wrwrwr);
draw(shift((-2.6097092102082557,1.0018165217625874))*xscale(0.07656252040410111)*yscale(0.07656252040410111)*arc((0,0),1,4.6482724472355805,13.48892752260185), linewidth(2) + wrwrwr);
draw(shift((-2.5892964855903644,1.0308440180397496))*xscale(0.052165292319906444)*yscale(0.052165292319906444)*arc((0,0),1,-1.6974259224121713,5.543047336484977), linewidth(2) + wrwrwr);
real implicitf5 (real x, real y) { return -5467.5244403743445+3051.5483510183285*y^1-1841.795530656712*y^2+395.14672450848167*y^3-91.67876382775592*y^4-7255.536320223792*x^1+2024.7179009537413*x^1*y^1-939.5257246444035*x^1*y^2-3823.0611290972947*x^2+395.14672450848167*x^2*y^1-183.35752765551183*x^2*y^2-939.5257246444035*x^3-91.67876382775592*x^4; }
draw(contour(implicitf5, (xmin,ymin), (xmax,ymax), new real[]{0}, 500), linewidth(2) + wrwrwr);
/* dots and labels */
dot((-2.610799442261206,1.078371279450788),dotstyle);
label("$O$", (-2.6029865194844883,1.0971208315102876), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/0/9/2/092060cb56af740942800cfcf037f9dbaf7de3c0.png)
This seems something really stupid
and funny indeed 
.
July 2020
April 2020