Circumradius of a perturbed equilateral triangle











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Consider an equilateral triangle $ABC$ with side lengths 1, on the picture with its circumcircle outlined. Its circumradius will be $1/sqrt{3}$.



enter image description here



Now imagine we allow each vertex to move within a disc of radius $rho$ centered at that vertex. We end up with a new triangle $A'B'C'$, where e.g. $A' in B(A,rho)$, the disc with center $A$ and radius $rho$.



enter image description here



The question is simple:




What is the maximum circumradius of the perturbed triangle $A'B'C'$?






According to Existence of Gibbsian point processes with geometry-dependent interactions (though I wouldn't recommend looking through the paper for any insights, as it deals with a completely different topic and gives no details on this problem) the maximum circumradius, for $rho leq sqrt 3 /6$, is
$$1/sqrt{3} + rho$$
Which is a very simple solution, but though it intuitively makes some sense to me, I can't really convince myself of it or prove it. It also leads to a secondary question




What happens at $rho = sqrt3/6$? How does the solution change for $rho > sqrt3 / 6$?




Intuitively what I think happens is that for small enough $rho$ the solution is still an equilateral triangle but after a certain point (probably $rho = sqrt 3/6$) this is no longer the case, as moving the points closer to collinearity will yield a greater circumradius, until finally at $rho = sqrt3/4$ the points can become collinear.










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    up vote
    6
    down vote

    favorite
    3












    Consider an equilateral triangle $ABC$ with side lengths 1, on the picture with its circumcircle outlined. Its circumradius will be $1/sqrt{3}$.



    enter image description here



    Now imagine we allow each vertex to move within a disc of radius $rho$ centered at that vertex. We end up with a new triangle $A'B'C'$, where e.g. $A' in B(A,rho)$, the disc with center $A$ and radius $rho$.



    enter image description here



    The question is simple:




    What is the maximum circumradius of the perturbed triangle $A'B'C'$?






    According to Existence of Gibbsian point processes with geometry-dependent interactions (though I wouldn't recommend looking through the paper for any insights, as it deals with a completely different topic and gives no details on this problem) the maximum circumradius, for $rho leq sqrt 3 /6$, is
    $$1/sqrt{3} + rho$$
    Which is a very simple solution, but though it intuitively makes some sense to me, I can't really convince myself of it or prove it. It also leads to a secondary question




    What happens at $rho = sqrt3/6$? How does the solution change for $rho > sqrt3 / 6$?




    Intuitively what I think happens is that for small enough $rho$ the solution is still an equilateral triangle but after a certain point (probably $rho = sqrt 3/6$) this is no longer the case, as moving the points closer to collinearity will yield a greater circumradius, until finally at $rho = sqrt3/4$ the points can become collinear.










    share|cite|improve this question


























      up vote
      6
      down vote

      favorite
      3









      up vote
      6
      down vote

      favorite
      3






      3





      Consider an equilateral triangle $ABC$ with side lengths 1, on the picture with its circumcircle outlined. Its circumradius will be $1/sqrt{3}$.



      enter image description here



      Now imagine we allow each vertex to move within a disc of radius $rho$ centered at that vertex. We end up with a new triangle $A'B'C'$, where e.g. $A' in B(A,rho)$, the disc with center $A$ and radius $rho$.



      enter image description here



      The question is simple:




      What is the maximum circumradius of the perturbed triangle $A'B'C'$?






      According to Existence of Gibbsian point processes with geometry-dependent interactions (though I wouldn't recommend looking through the paper for any insights, as it deals with a completely different topic and gives no details on this problem) the maximum circumradius, for $rho leq sqrt 3 /6$, is
      $$1/sqrt{3} + rho$$
      Which is a very simple solution, but though it intuitively makes some sense to me, I can't really convince myself of it or prove it. It also leads to a secondary question




      What happens at $rho = sqrt3/6$? How does the solution change for $rho > sqrt3 / 6$?




      Intuitively what I think happens is that for small enough $rho$ the solution is still an equilateral triangle but after a certain point (probably $rho = sqrt 3/6$) this is no longer the case, as moving the points closer to collinearity will yield a greater circumradius, until finally at $rho = sqrt3/4$ the points can become collinear.










      share|cite|improve this question















      Consider an equilateral triangle $ABC$ with side lengths 1, on the picture with its circumcircle outlined. Its circumradius will be $1/sqrt{3}$.



      enter image description here



      Now imagine we allow each vertex to move within a disc of radius $rho$ centered at that vertex. We end up with a new triangle $A'B'C'$, where e.g. $A' in B(A,rho)$, the disc with center $A$ and radius $rho$.



      enter image description here



      The question is simple:




      What is the maximum circumradius of the perturbed triangle $A'B'C'$?






      According to Existence of Gibbsian point processes with geometry-dependent interactions (though I wouldn't recommend looking through the paper for any insights, as it deals with a completely different topic and gives no details on this problem) the maximum circumradius, for $rho leq sqrt 3 /6$, is
      $$1/sqrt{3} + rho$$
      Which is a very simple solution, but though it intuitively makes some sense to me, I can't really convince myself of it or prove it. It also leads to a secondary question




      What happens at $rho = sqrt3/6$? How does the solution change for $rho > sqrt3 / 6$?




      Intuitively what I think happens is that for small enough $rho$ the solution is still an equilateral triangle but after a certain point (probably $rho = sqrt 3/6$) this is no longer the case, as moving the points closer to collinearity will yield a greater circumradius, until finally at $rho = sqrt3/4$ the points can become collinear.







      geometry trigonometry euclidean-geometry triangle






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      edited Nov 17 at 13:55

























      asked Nov 17 at 11:02









      Dahn Jahn

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          Why the circumradius is maximised when all three perturbed points are on the boundary of the small circles

          Suppose the small circles are disjoint, there is no line intersecting all three circles, points $A'$ and $B'$ are fixed and $C'$ is not on the boundary of its circle (i.e. it has a neighbourhood within the circle around $C$). The circumradius is a differentiable, non-constant function of the coordinates of $C'$. Thus there must be some perturbation of $C'$, within the aforementioned neighbourhood, that increases the circumradius – there are no local extrema to impede this modification because the level sets of the circumradius function can only be as small as the circle with diameter $A'B'$, whereas arbitrarily small level sets can be drawn around a local extremum.



          Thus the circumradius can only be maximised when $A',B',C'$ are all on the boundaries of their circles, at which point the circumcircle is tangent to all three small circles.





          Now the problem boils down to determining which of two such circles – $Gamma_3$ internally tangent to all three small circles, or $Gamma_2$ internally tangent to only two – is larger:





          (It can be easily shown that the corresponding $Gamma_1$ and $Gamma_0$ are never larger than $Gamma_3$; we choose the vertices of $triangle A'B'C'$ as the points of tangency.) It is easy to show that $r(Gamma_3)=frac1{sqrt3}+rho$, since it is concentric with the original equilateral triangle, which has circumradius $frac1{sqrt3}$. To determine the radius of $Gamma_2$, we draw another diagram:





          From this we derive the equation
          $$frac{sqrt3}2-rho+k-rho=sqrt{frac14+k^2}$$
          $$(k+sqrt3/2-2rho)^2=k^2+(sqrt3-4rho)k+(sqrt3/2-2rho)^2=frac14+k^2$$
          $$(sqrt3-4rho)k+frac34-2sqrt3rho+4rho^2=frac14$$
          $$k=frac{-frac12+2sqrt3rho-4rho^2}{sqrt3-4rho}$$
          $$r(Gamma_2)=frac{-frac12+2sqrt3rho-4rho^2}{sqrt3-4rho}+frac{sqrt3}2-rho=frac{1-sqrt3rho}{sqrt3-4rho}$$
          Now where is this greater than $r(Gamma_3)$?
          $$frac{1-sqrt3rho}{sqrt3-4rho}=frac1{sqrt3}+rho$$
          $$1-sqrt3rho=(1/sqrt3+rho)(sqrt3-4rho)=1-4/sqrt3rho+sqrt3rho-4rho^2$$
          $$4sqrt3rho^2-2rho=0qquadrho=frac{sqrt3}6$$
          Thus, above this critical value $Gamma_2$ is larger and vice versa.



          The vertical asymptote of $r(Gamma_2)=frac{1-sqrt3rho}{sqrt3-4rho}$ comes at $rho=frac{sqrt3}4$; at and above this value we can choose three collinear points.



          Thus the maximum circumradius of $triangle A'B'C'$ is
          $$begin{cases}frac1{sqrt3}+rho&rho<frac{sqrt3}6\
          frac{1-sqrt3rho}{sqrt3-4rho}&frac{sqrt3}6lerho<frac{sqrt3}4\
          infty&rhogefrac{sqrt3}4end{cases}$$






          share|cite|improve this answer























          • Thank you, I'm almost there. I don't quite understand the definition of $Gamma_i$s. For instance, is $Gamma_2$ any circle that is internally tangent to two of the circles (and can cross, rather than be tangent to, the third)? If that's the case, then surely I can make e.g. $r(Gamma_0)$ arbitrarily close to $r(Gamma_i)$ for any $i$.
            – Dahn Jahn
            Nov 17 at 13:50










          • Ah, this lead me to a mistake I made - I mistook a disc for a circle. I edited the question accordingly. I am guessing the solution doesn't change, as the points maximizing the circumradius will still lie on the circles, but is there a simple argument as to why that's true? Sorry for the confusion.
            – Dahn Jahn
            Nov 17 at 13:58












          • I see how this shows that (Solution is internally tangent to 2) $Rightarrow$ (Solution is tangent to all 3), but I don't understand how this shows that the maximizing solution cannot have points on the interior in general.
            – Dahn Jahn
            Nov 17 at 14:18










          • I am having trouble understanding this. Could you elaborate, perhaps even as a part of the post for completeness? Thank you.
            – Dahn Jahn
            Nov 17 at 14:44






          • 1




            @DahnJahn The level sets of the circumradius function are circles themselves, passing through $A',B'$. These circles cannot get smaller than the diameter $A'B'$. If there was a local extremum, we could draw an arbitrarily small level set around it, but we just showed that the level sets are lower-bounded in size, so there is no local extremum.
            – Parcly Taxel
            Nov 17 at 18:26











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          Why the circumradius is maximised when all three perturbed points are on the boundary of the small circles

          Suppose the small circles are disjoint, there is no line intersecting all three circles, points $A'$ and $B'$ are fixed and $C'$ is not on the boundary of its circle (i.e. it has a neighbourhood within the circle around $C$). The circumradius is a differentiable, non-constant function of the coordinates of $C'$. Thus there must be some perturbation of $C'$, within the aforementioned neighbourhood, that increases the circumradius – there are no local extrema to impede this modification because the level sets of the circumradius function can only be as small as the circle with diameter $A'B'$, whereas arbitrarily small level sets can be drawn around a local extremum.



          Thus the circumradius can only be maximised when $A',B',C'$ are all on the boundaries of their circles, at which point the circumcircle is tangent to all three small circles.





          Now the problem boils down to determining which of two such circles – $Gamma_3$ internally tangent to all three small circles, or $Gamma_2$ internally tangent to only two – is larger:





          (It can be easily shown that the corresponding $Gamma_1$ and $Gamma_0$ are never larger than $Gamma_3$; we choose the vertices of $triangle A'B'C'$ as the points of tangency.) It is easy to show that $r(Gamma_3)=frac1{sqrt3}+rho$, since it is concentric with the original equilateral triangle, which has circumradius $frac1{sqrt3}$. To determine the radius of $Gamma_2$, we draw another diagram:





          From this we derive the equation
          $$frac{sqrt3}2-rho+k-rho=sqrt{frac14+k^2}$$
          $$(k+sqrt3/2-2rho)^2=k^2+(sqrt3-4rho)k+(sqrt3/2-2rho)^2=frac14+k^2$$
          $$(sqrt3-4rho)k+frac34-2sqrt3rho+4rho^2=frac14$$
          $$k=frac{-frac12+2sqrt3rho-4rho^2}{sqrt3-4rho}$$
          $$r(Gamma_2)=frac{-frac12+2sqrt3rho-4rho^2}{sqrt3-4rho}+frac{sqrt3}2-rho=frac{1-sqrt3rho}{sqrt3-4rho}$$
          Now where is this greater than $r(Gamma_3)$?
          $$frac{1-sqrt3rho}{sqrt3-4rho}=frac1{sqrt3}+rho$$
          $$1-sqrt3rho=(1/sqrt3+rho)(sqrt3-4rho)=1-4/sqrt3rho+sqrt3rho-4rho^2$$
          $$4sqrt3rho^2-2rho=0qquadrho=frac{sqrt3}6$$
          Thus, above this critical value $Gamma_2$ is larger and vice versa.



          The vertical asymptote of $r(Gamma_2)=frac{1-sqrt3rho}{sqrt3-4rho}$ comes at $rho=frac{sqrt3}4$; at and above this value we can choose three collinear points.



          Thus the maximum circumradius of $triangle A'B'C'$ is
          $$begin{cases}frac1{sqrt3}+rho&rho<frac{sqrt3}6\
          frac{1-sqrt3rho}{sqrt3-4rho}&frac{sqrt3}6lerho<frac{sqrt3}4\
          infty&rhogefrac{sqrt3}4end{cases}$$






          share|cite|improve this answer























          • Thank you, I'm almost there. I don't quite understand the definition of $Gamma_i$s. For instance, is $Gamma_2$ any circle that is internally tangent to two of the circles (and can cross, rather than be tangent to, the third)? If that's the case, then surely I can make e.g. $r(Gamma_0)$ arbitrarily close to $r(Gamma_i)$ for any $i$.
            – Dahn Jahn
            Nov 17 at 13:50










          • Ah, this lead me to a mistake I made - I mistook a disc for a circle. I edited the question accordingly. I am guessing the solution doesn't change, as the points maximizing the circumradius will still lie on the circles, but is there a simple argument as to why that's true? Sorry for the confusion.
            – Dahn Jahn
            Nov 17 at 13:58












          • I see how this shows that (Solution is internally tangent to 2) $Rightarrow$ (Solution is tangent to all 3), but I don't understand how this shows that the maximizing solution cannot have points on the interior in general.
            – Dahn Jahn
            Nov 17 at 14:18










          • I am having trouble understanding this. Could you elaborate, perhaps even as a part of the post for completeness? Thank you.
            – Dahn Jahn
            Nov 17 at 14:44






          • 1




            @DahnJahn The level sets of the circumradius function are circles themselves, passing through $A',B'$. These circles cannot get smaller than the diameter $A'B'$. If there was a local extremum, we could draw an arbitrarily small level set around it, but we just showed that the level sets are lower-bounded in size, so there is no local extremum.
            – Parcly Taxel
            Nov 17 at 18:26















          up vote
          2
          down vote



          accepted










          Why the circumradius is maximised when all three perturbed points are on the boundary of the small circles

          Suppose the small circles are disjoint, there is no line intersecting all three circles, points $A'$ and $B'$ are fixed and $C'$ is not on the boundary of its circle (i.e. it has a neighbourhood within the circle around $C$). The circumradius is a differentiable, non-constant function of the coordinates of $C'$. Thus there must be some perturbation of $C'$, within the aforementioned neighbourhood, that increases the circumradius – there are no local extrema to impede this modification because the level sets of the circumradius function can only be as small as the circle with diameter $A'B'$, whereas arbitrarily small level sets can be drawn around a local extremum.



          Thus the circumradius can only be maximised when $A',B',C'$ are all on the boundaries of their circles, at which point the circumcircle is tangent to all three small circles.





          Now the problem boils down to determining which of two such circles – $Gamma_3$ internally tangent to all three small circles, or $Gamma_2$ internally tangent to only two – is larger:





          (It can be easily shown that the corresponding $Gamma_1$ and $Gamma_0$ are never larger than $Gamma_3$; we choose the vertices of $triangle A'B'C'$ as the points of tangency.) It is easy to show that $r(Gamma_3)=frac1{sqrt3}+rho$, since it is concentric with the original equilateral triangle, which has circumradius $frac1{sqrt3}$. To determine the radius of $Gamma_2$, we draw another diagram:





          From this we derive the equation
          $$frac{sqrt3}2-rho+k-rho=sqrt{frac14+k^2}$$
          $$(k+sqrt3/2-2rho)^2=k^2+(sqrt3-4rho)k+(sqrt3/2-2rho)^2=frac14+k^2$$
          $$(sqrt3-4rho)k+frac34-2sqrt3rho+4rho^2=frac14$$
          $$k=frac{-frac12+2sqrt3rho-4rho^2}{sqrt3-4rho}$$
          $$r(Gamma_2)=frac{-frac12+2sqrt3rho-4rho^2}{sqrt3-4rho}+frac{sqrt3}2-rho=frac{1-sqrt3rho}{sqrt3-4rho}$$
          Now where is this greater than $r(Gamma_3)$?
          $$frac{1-sqrt3rho}{sqrt3-4rho}=frac1{sqrt3}+rho$$
          $$1-sqrt3rho=(1/sqrt3+rho)(sqrt3-4rho)=1-4/sqrt3rho+sqrt3rho-4rho^2$$
          $$4sqrt3rho^2-2rho=0qquadrho=frac{sqrt3}6$$
          Thus, above this critical value $Gamma_2$ is larger and vice versa.



          The vertical asymptote of $r(Gamma_2)=frac{1-sqrt3rho}{sqrt3-4rho}$ comes at $rho=frac{sqrt3}4$; at and above this value we can choose three collinear points.



          Thus the maximum circumradius of $triangle A'B'C'$ is
          $$begin{cases}frac1{sqrt3}+rho&rho<frac{sqrt3}6\
          frac{1-sqrt3rho}{sqrt3-4rho}&frac{sqrt3}6lerho<frac{sqrt3}4\
          infty&rhogefrac{sqrt3}4end{cases}$$






          share|cite|improve this answer























          • Thank you, I'm almost there. I don't quite understand the definition of $Gamma_i$s. For instance, is $Gamma_2$ any circle that is internally tangent to two of the circles (and can cross, rather than be tangent to, the third)? If that's the case, then surely I can make e.g. $r(Gamma_0)$ arbitrarily close to $r(Gamma_i)$ for any $i$.
            – Dahn Jahn
            Nov 17 at 13:50










          • Ah, this lead me to a mistake I made - I mistook a disc for a circle. I edited the question accordingly. I am guessing the solution doesn't change, as the points maximizing the circumradius will still lie on the circles, but is there a simple argument as to why that's true? Sorry for the confusion.
            – Dahn Jahn
            Nov 17 at 13:58












          • I see how this shows that (Solution is internally tangent to 2) $Rightarrow$ (Solution is tangent to all 3), but I don't understand how this shows that the maximizing solution cannot have points on the interior in general.
            – Dahn Jahn
            Nov 17 at 14:18










          • I am having trouble understanding this. Could you elaborate, perhaps even as a part of the post for completeness? Thank you.
            – Dahn Jahn
            Nov 17 at 14:44






          • 1




            @DahnJahn The level sets of the circumradius function are circles themselves, passing through $A',B'$. These circles cannot get smaller than the diameter $A'B'$. If there was a local extremum, we could draw an arbitrarily small level set around it, but we just showed that the level sets are lower-bounded in size, so there is no local extremum.
            – Parcly Taxel
            Nov 17 at 18:26













          up vote
          2
          down vote



          accepted







          up vote
          2
          down vote



          accepted






          Why the circumradius is maximised when all three perturbed points are on the boundary of the small circles

          Suppose the small circles are disjoint, there is no line intersecting all three circles, points $A'$ and $B'$ are fixed and $C'$ is not on the boundary of its circle (i.e. it has a neighbourhood within the circle around $C$). The circumradius is a differentiable, non-constant function of the coordinates of $C'$. Thus there must be some perturbation of $C'$, within the aforementioned neighbourhood, that increases the circumradius – there are no local extrema to impede this modification because the level sets of the circumradius function can only be as small as the circle with diameter $A'B'$, whereas arbitrarily small level sets can be drawn around a local extremum.



          Thus the circumradius can only be maximised when $A',B',C'$ are all on the boundaries of their circles, at which point the circumcircle is tangent to all three small circles.





          Now the problem boils down to determining which of two such circles – $Gamma_3$ internally tangent to all three small circles, or $Gamma_2$ internally tangent to only two – is larger:





          (It can be easily shown that the corresponding $Gamma_1$ and $Gamma_0$ are never larger than $Gamma_3$; we choose the vertices of $triangle A'B'C'$ as the points of tangency.) It is easy to show that $r(Gamma_3)=frac1{sqrt3}+rho$, since it is concentric with the original equilateral triangle, which has circumradius $frac1{sqrt3}$. To determine the radius of $Gamma_2$, we draw another diagram:





          From this we derive the equation
          $$frac{sqrt3}2-rho+k-rho=sqrt{frac14+k^2}$$
          $$(k+sqrt3/2-2rho)^2=k^2+(sqrt3-4rho)k+(sqrt3/2-2rho)^2=frac14+k^2$$
          $$(sqrt3-4rho)k+frac34-2sqrt3rho+4rho^2=frac14$$
          $$k=frac{-frac12+2sqrt3rho-4rho^2}{sqrt3-4rho}$$
          $$r(Gamma_2)=frac{-frac12+2sqrt3rho-4rho^2}{sqrt3-4rho}+frac{sqrt3}2-rho=frac{1-sqrt3rho}{sqrt3-4rho}$$
          Now where is this greater than $r(Gamma_3)$?
          $$frac{1-sqrt3rho}{sqrt3-4rho}=frac1{sqrt3}+rho$$
          $$1-sqrt3rho=(1/sqrt3+rho)(sqrt3-4rho)=1-4/sqrt3rho+sqrt3rho-4rho^2$$
          $$4sqrt3rho^2-2rho=0qquadrho=frac{sqrt3}6$$
          Thus, above this critical value $Gamma_2$ is larger and vice versa.



          The vertical asymptote of $r(Gamma_2)=frac{1-sqrt3rho}{sqrt3-4rho}$ comes at $rho=frac{sqrt3}4$; at and above this value we can choose three collinear points.



          Thus the maximum circumradius of $triangle A'B'C'$ is
          $$begin{cases}frac1{sqrt3}+rho&rho<frac{sqrt3}6\
          frac{1-sqrt3rho}{sqrt3-4rho}&frac{sqrt3}6lerho<frac{sqrt3}4\
          infty&rhogefrac{sqrt3}4end{cases}$$






          share|cite|improve this answer














          Why the circumradius is maximised when all three perturbed points are on the boundary of the small circles

          Suppose the small circles are disjoint, there is no line intersecting all three circles, points $A'$ and $B'$ are fixed and $C'$ is not on the boundary of its circle (i.e. it has a neighbourhood within the circle around $C$). The circumradius is a differentiable, non-constant function of the coordinates of $C'$. Thus there must be some perturbation of $C'$, within the aforementioned neighbourhood, that increases the circumradius – there are no local extrema to impede this modification because the level sets of the circumradius function can only be as small as the circle with diameter $A'B'$, whereas arbitrarily small level sets can be drawn around a local extremum.



          Thus the circumradius can only be maximised when $A',B',C'$ are all on the boundaries of their circles, at which point the circumcircle is tangent to all three small circles.





          Now the problem boils down to determining which of two such circles – $Gamma_3$ internally tangent to all three small circles, or $Gamma_2$ internally tangent to only two – is larger:





          (It can be easily shown that the corresponding $Gamma_1$ and $Gamma_0$ are never larger than $Gamma_3$; we choose the vertices of $triangle A'B'C'$ as the points of tangency.) It is easy to show that $r(Gamma_3)=frac1{sqrt3}+rho$, since it is concentric with the original equilateral triangle, which has circumradius $frac1{sqrt3}$. To determine the radius of $Gamma_2$, we draw another diagram:





          From this we derive the equation
          $$frac{sqrt3}2-rho+k-rho=sqrt{frac14+k^2}$$
          $$(k+sqrt3/2-2rho)^2=k^2+(sqrt3-4rho)k+(sqrt3/2-2rho)^2=frac14+k^2$$
          $$(sqrt3-4rho)k+frac34-2sqrt3rho+4rho^2=frac14$$
          $$k=frac{-frac12+2sqrt3rho-4rho^2}{sqrt3-4rho}$$
          $$r(Gamma_2)=frac{-frac12+2sqrt3rho-4rho^2}{sqrt3-4rho}+frac{sqrt3}2-rho=frac{1-sqrt3rho}{sqrt3-4rho}$$
          Now where is this greater than $r(Gamma_3)$?
          $$frac{1-sqrt3rho}{sqrt3-4rho}=frac1{sqrt3}+rho$$
          $$1-sqrt3rho=(1/sqrt3+rho)(sqrt3-4rho)=1-4/sqrt3rho+sqrt3rho-4rho^2$$
          $$4sqrt3rho^2-2rho=0qquadrho=frac{sqrt3}6$$
          Thus, above this critical value $Gamma_2$ is larger and vice versa.



          The vertical asymptote of $r(Gamma_2)=frac{1-sqrt3rho}{sqrt3-4rho}$ comes at $rho=frac{sqrt3}4$; at and above this value we can choose three collinear points.



          Thus the maximum circumradius of $triangle A'B'C'$ is
          $$begin{cases}frac1{sqrt3}+rho&rho<frac{sqrt3}6\
          frac{1-sqrt3rho}{sqrt3-4rho}&frac{sqrt3}6lerho<frac{sqrt3}4\
          infty&rhogefrac{sqrt3}4end{cases}$$







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Nov 17 at 18:33

























          answered Nov 17 at 11:42









          Parcly Taxel

          41k137198




          41k137198












          • Thank you, I'm almost there. I don't quite understand the definition of $Gamma_i$s. For instance, is $Gamma_2$ any circle that is internally tangent to two of the circles (and can cross, rather than be tangent to, the third)? If that's the case, then surely I can make e.g. $r(Gamma_0)$ arbitrarily close to $r(Gamma_i)$ for any $i$.
            – Dahn Jahn
            Nov 17 at 13:50










          • Ah, this lead me to a mistake I made - I mistook a disc for a circle. I edited the question accordingly. I am guessing the solution doesn't change, as the points maximizing the circumradius will still lie on the circles, but is there a simple argument as to why that's true? Sorry for the confusion.
            – Dahn Jahn
            Nov 17 at 13:58












          • I see how this shows that (Solution is internally tangent to 2) $Rightarrow$ (Solution is tangent to all 3), but I don't understand how this shows that the maximizing solution cannot have points on the interior in general.
            – Dahn Jahn
            Nov 17 at 14:18










          • I am having trouble understanding this. Could you elaborate, perhaps even as a part of the post for completeness? Thank you.
            – Dahn Jahn
            Nov 17 at 14:44






          • 1




            @DahnJahn The level sets of the circumradius function are circles themselves, passing through $A',B'$. These circles cannot get smaller than the diameter $A'B'$. If there was a local extremum, we could draw an arbitrarily small level set around it, but we just showed that the level sets are lower-bounded in size, so there is no local extremum.
            – Parcly Taxel
            Nov 17 at 18:26


















          • Thank you, I'm almost there. I don't quite understand the definition of $Gamma_i$s. For instance, is $Gamma_2$ any circle that is internally tangent to two of the circles (and can cross, rather than be tangent to, the third)? If that's the case, then surely I can make e.g. $r(Gamma_0)$ arbitrarily close to $r(Gamma_i)$ for any $i$.
            – Dahn Jahn
            Nov 17 at 13:50










          • Ah, this lead me to a mistake I made - I mistook a disc for a circle. I edited the question accordingly. I am guessing the solution doesn't change, as the points maximizing the circumradius will still lie on the circles, but is there a simple argument as to why that's true? Sorry for the confusion.
            – Dahn Jahn
            Nov 17 at 13:58












          • I see how this shows that (Solution is internally tangent to 2) $Rightarrow$ (Solution is tangent to all 3), but I don't understand how this shows that the maximizing solution cannot have points on the interior in general.
            – Dahn Jahn
            Nov 17 at 14:18










          • I am having trouble understanding this. Could you elaborate, perhaps even as a part of the post for completeness? Thank you.
            – Dahn Jahn
            Nov 17 at 14:44






          • 1




            @DahnJahn The level sets of the circumradius function are circles themselves, passing through $A',B'$. These circles cannot get smaller than the diameter $A'B'$. If there was a local extremum, we could draw an arbitrarily small level set around it, but we just showed that the level sets are lower-bounded in size, so there is no local extremum.
            – Parcly Taxel
            Nov 17 at 18:26
















          Thank you, I'm almost there. I don't quite understand the definition of $Gamma_i$s. For instance, is $Gamma_2$ any circle that is internally tangent to two of the circles (and can cross, rather than be tangent to, the third)? If that's the case, then surely I can make e.g. $r(Gamma_0)$ arbitrarily close to $r(Gamma_i)$ for any $i$.
          – Dahn Jahn
          Nov 17 at 13:50




          Thank you, I'm almost there. I don't quite understand the definition of $Gamma_i$s. For instance, is $Gamma_2$ any circle that is internally tangent to two of the circles (and can cross, rather than be tangent to, the third)? If that's the case, then surely I can make e.g. $r(Gamma_0)$ arbitrarily close to $r(Gamma_i)$ for any $i$.
          – Dahn Jahn
          Nov 17 at 13:50












          Ah, this lead me to a mistake I made - I mistook a disc for a circle. I edited the question accordingly. I am guessing the solution doesn't change, as the points maximizing the circumradius will still lie on the circles, but is there a simple argument as to why that's true? Sorry for the confusion.
          – Dahn Jahn
          Nov 17 at 13:58






          Ah, this lead me to a mistake I made - I mistook a disc for a circle. I edited the question accordingly. I am guessing the solution doesn't change, as the points maximizing the circumradius will still lie on the circles, but is there a simple argument as to why that's true? Sorry for the confusion.
          – Dahn Jahn
          Nov 17 at 13:58














          I see how this shows that (Solution is internally tangent to 2) $Rightarrow$ (Solution is tangent to all 3), but I don't understand how this shows that the maximizing solution cannot have points on the interior in general.
          – Dahn Jahn
          Nov 17 at 14:18




          I see how this shows that (Solution is internally tangent to 2) $Rightarrow$ (Solution is tangent to all 3), but I don't understand how this shows that the maximizing solution cannot have points on the interior in general.
          – Dahn Jahn
          Nov 17 at 14:18












          I am having trouble understanding this. Could you elaborate, perhaps even as a part of the post for completeness? Thank you.
          – Dahn Jahn
          Nov 17 at 14:44




          I am having trouble understanding this. Could you elaborate, perhaps even as a part of the post for completeness? Thank you.
          – Dahn Jahn
          Nov 17 at 14:44




          1




          1




          @DahnJahn The level sets of the circumradius function are circles themselves, passing through $A',B'$. These circles cannot get smaller than the diameter $A'B'$. If there was a local extremum, we could draw an arbitrarily small level set around it, but we just showed that the level sets are lower-bounded in size, so there is no local extremum.
          – Parcly Taxel
          Nov 17 at 18:26




          @DahnJahn The level sets of the circumradius function are circles themselves, passing through $A',B'$. These circles cannot get smaller than the diameter $A'B'$. If there was a local extremum, we could draw an arbitrarily small level set around it, but we just showed that the level sets are lower-bounded in size, so there is no local extremum.
          – Parcly Taxel
          Nov 17 at 18:26


















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