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Overlapping circles covering polygon


The locus of centre of circle tangent to two given circlesArea of the Limiting PolygonThree circles having centres on the three sides of a triangleTwo conics from six Thebault circles of a triangleTangent circles and finding diametersNew Golden Ratio Construct: which one of my constructs is superior/simplest--squares & circles or just circles?Pencils of CirclesIntersection of two circles dividing lune in given ratioMaximum number of circles tangent to two concentric circlesFinding the radius of incircle of $triangle ABC$ where each of three other circles is mutually tangent to two sides of the triangle respectively













10












$begingroup$


While working in GeoGebra I noticed something odd. I had a triangle with a point inside and the point was connected to each of the vertices. For each vertice I had drawn the circle passing through the vertice and the point, with the connection being the circle's diameter (see picture below).



enter image description here



What I noticed is that the overlapping circles completely covered the triangle. Further experimentation showed this was also the case if the point was outside the triangle (see below).



enter image description here



Still more experimentation appears to show this is the case for any polygon, simple or not:



enter image description here



Is this observation true or did GeoGebra lead me astray? I couldn't immediately find the result via Google.










share|cite|improve this question









$endgroup$











  • $begingroup$
    There is another remarkable detail that the intersection point of any pair of circles distinct from the red point lies on the line connecting two vertices. And this fact is quite easy to prove.
    $endgroup$
    – user
    yesterday











  • $begingroup$
    @user I noticed that too. Could be a starting point for proving the above is true (if it is true).
    $endgroup$
    – Jens
    yesterday










  • $begingroup$
    Yes this can be the starting point. In the case of internal point of the triangle it is practically evident. In the case of polygons it suffices to consider its convex hull.
    $endgroup$
    – user
    yesterday







  • 1




    $begingroup$
    Given any polygon $mathcalP = ABCcdots F$ and a point $X$. Consider triangles formed by $X$ and an edge of $mathcalP$. Let's say triangle $XAB$. Let $Y$ be the foot of the altitude through $X$ on $AB$. Split $XAB$ into two right angled triangles $XAY$ and $XYB$. These two triangles will be covered by the two circles with $XA$ and $XB$ as diameters....
    $endgroup$
    – achille hui
    yesterday











  • $begingroup$
    For future reference, "vertice" is not a word. You can talk about each vertex of a triangle. Vertices is the plural of vertex. It's not the usual way of forming a plural in English, because it isn't English--it's Latin. If you were to speak of multiple vertexes, however, I wouldn't complain. That's a legitimate alternative plural formation in English without the Latin pretentiousness.
    $endgroup$
    – David K
    23 hours ago
















10












$begingroup$


While working in GeoGebra I noticed something odd. I had a triangle with a point inside and the point was connected to each of the vertices. For each vertice I had drawn the circle passing through the vertice and the point, with the connection being the circle's diameter (see picture below).



enter image description here



What I noticed is that the overlapping circles completely covered the triangle. Further experimentation showed this was also the case if the point was outside the triangle (see below).



enter image description here



Still more experimentation appears to show this is the case for any polygon, simple or not:



enter image description here



Is this observation true or did GeoGebra lead me astray? I couldn't immediately find the result via Google.










share|cite|improve this question









$endgroup$











  • $begingroup$
    There is another remarkable detail that the intersection point of any pair of circles distinct from the red point lies on the line connecting two vertices. And this fact is quite easy to prove.
    $endgroup$
    – user
    yesterday











  • $begingroup$
    @user I noticed that too. Could be a starting point for proving the above is true (if it is true).
    $endgroup$
    – Jens
    yesterday










  • $begingroup$
    Yes this can be the starting point. In the case of internal point of the triangle it is practically evident. In the case of polygons it suffices to consider its convex hull.
    $endgroup$
    – user
    yesterday







  • 1




    $begingroup$
    Given any polygon $mathcalP = ABCcdots F$ and a point $X$. Consider triangles formed by $X$ and an edge of $mathcalP$. Let's say triangle $XAB$. Let $Y$ be the foot of the altitude through $X$ on $AB$. Split $XAB$ into two right angled triangles $XAY$ and $XYB$. These two triangles will be covered by the two circles with $XA$ and $XB$ as diameters....
    $endgroup$
    – achille hui
    yesterday











  • $begingroup$
    For future reference, "vertice" is not a word. You can talk about each vertex of a triangle. Vertices is the plural of vertex. It's not the usual way of forming a plural in English, because it isn't English--it's Latin. If you were to speak of multiple vertexes, however, I wouldn't complain. That's a legitimate alternative plural formation in English without the Latin pretentiousness.
    $endgroup$
    – David K
    23 hours ago














10












10








10





$begingroup$


While working in GeoGebra I noticed something odd. I had a triangle with a point inside and the point was connected to each of the vertices. For each vertice I had drawn the circle passing through the vertice and the point, with the connection being the circle's diameter (see picture below).



enter image description here



What I noticed is that the overlapping circles completely covered the triangle. Further experimentation showed this was also the case if the point was outside the triangle (see below).



enter image description here



Still more experimentation appears to show this is the case for any polygon, simple or not:



enter image description here



Is this observation true or did GeoGebra lead me astray? I couldn't immediately find the result via Google.










share|cite|improve this question









$endgroup$




While working in GeoGebra I noticed something odd. I had a triangle with a point inside and the point was connected to each of the vertices. For each vertice I had drawn the circle passing through the vertice and the point, with the connection being the circle's diameter (see picture below).



enter image description here



What I noticed is that the overlapping circles completely covered the triangle. Further experimentation showed this was also the case if the point was outside the triangle (see below).



enter image description here



Still more experimentation appears to show this is the case for any polygon, simple or not:



enter image description here



Is this observation true or did GeoGebra lead me astray? I couldn't immediately find the result via Google.







geometry recreational-mathematics






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked yesterday









JensJens

4,02721032




4,02721032











  • $begingroup$
    There is another remarkable detail that the intersection point of any pair of circles distinct from the red point lies on the line connecting two vertices. And this fact is quite easy to prove.
    $endgroup$
    – user
    yesterday











  • $begingroup$
    @user I noticed that too. Could be a starting point for proving the above is true (if it is true).
    $endgroup$
    – Jens
    yesterday










  • $begingroup$
    Yes this can be the starting point. In the case of internal point of the triangle it is practically evident. In the case of polygons it suffices to consider its convex hull.
    $endgroup$
    – user
    yesterday







  • 1




    $begingroup$
    Given any polygon $mathcalP = ABCcdots F$ and a point $X$. Consider triangles formed by $X$ and an edge of $mathcalP$. Let's say triangle $XAB$. Let $Y$ be the foot of the altitude through $X$ on $AB$. Split $XAB$ into two right angled triangles $XAY$ and $XYB$. These two triangles will be covered by the two circles with $XA$ and $XB$ as diameters....
    $endgroup$
    – achille hui
    yesterday











  • $begingroup$
    For future reference, "vertice" is not a word. You can talk about each vertex of a triangle. Vertices is the plural of vertex. It's not the usual way of forming a plural in English, because it isn't English--it's Latin. If you were to speak of multiple vertexes, however, I wouldn't complain. That's a legitimate alternative plural formation in English without the Latin pretentiousness.
    $endgroup$
    – David K
    23 hours ago

















  • $begingroup$
    There is another remarkable detail that the intersection point of any pair of circles distinct from the red point lies on the line connecting two vertices. And this fact is quite easy to prove.
    $endgroup$
    – user
    yesterday











  • $begingroup$
    @user I noticed that too. Could be a starting point for proving the above is true (if it is true).
    $endgroup$
    – Jens
    yesterday










  • $begingroup$
    Yes this can be the starting point. In the case of internal point of the triangle it is practically evident. In the case of polygons it suffices to consider its convex hull.
    $endgroup$
    – user
    yesterday







  • 1




    $begingroup$
    Given any polygon $mathcalP = ABCcdots F$ and a point $X$. Consider triangles formed by $X$ and an edge of $mathcalP$. Let's say triangle $XAB$. Let $Y$ be the foot of the altitude through $X$ on $AB$. Split $XAB$ into two right angled triangles $XAY$ and $XYB$. These two triangles will be covered by the two circles with $XA$ and $XB$ as diameters....
    $endgroup$
    – achille hui
    yesterday











  • $begingroup$
    For future reference, "vertice" is not a word. You can talk about each vertex of a triangle. Vertices is the plural of vertex. It's not the usual way of forming a plural in English, because it isn't English--it's Latin. If you were to speak of multiple vertexes, however, I wouldn't complain. That's a legitimate alternative plural formation in English without the Latin pretentiousness.
    $endgroup$
    – David K
    23 hours ago
















$begingroup$
There is another remarkable detail that the intersection point of any pair of circles distinct from the red point lies on the line connecting two vertices. And this fact is quite easy to prove.
$endgroup$
– user
yesterday





$begingroup$
There is another remarkable detail that the intersection point of any pair of circles distinct from the red point lies on the line connecting two vertices. And this fact is quite easy to prove.
$endgroup$
– user
yesterday













$begingroup$
@user I noticed that too. Could be a starting point for proving the above is true (if it is true).
$endgroup$
– Jens
yesterday




$begingroup$
@user I noticed that too. Could be a starting point for proving the above is true (if it is true).
$endgroup$
– Jens
yesterday












$begingroup$
Yes this can be the starting point. In the case of internal point of the triangle it is practically evident. In the case of polygons it suffices to consider its convex hull.
$endgroup$
– user
yesterday





$begingroup$
Yes this can be the starting point. In the case of internal point of the triangle it is practically evident. In the case of polygons it suffices to consider its convex hull.
$endgroup$
– user
yesterday





1




1




$begingroup$
Given any polygon $mathcalP = ABCcdots F$ and a point $X$. Consider triangles formed by $X$ and an edge of $mathcalP$. Let's say triangle $XAB$. Let $Y$ be the foot of the altitude through $X$ on $AB$. Split $XAB$ into two right angled triangles $XAY$ and $XYB$. These two triangles will be covered by the two circles with $XA$ and $XB$ as diameters....
$endgroup$
– achille hui
yesterday





$begingroup$
Given any polygon $mathcalP = ABCcdots F$ and a point $X$. Consider triangles formed by $X$ and an edge of $mathcalP$. Let's say triangle $XAB$. Let $Y$ be the foot of the altitude through $X$ on $AB$. Split $XAB$ into two right angled triangles $XAY$ and $XYB$. These two triangles will be covered by the two circles with $XA$ and $XB$ as diameters....
$endgroup$
– achille hui
yesterday













$begingroup$
For future reference, "vertice" is not a word. You can talk about each vertex of a triangle. Vertices is the plural of vertex. It's not the usual way of forming a plural in English, because it isn't English--it's Latin. If you were to speak of multiple vertexes, however, I wouldn't complain. That's a legitimate alternative plural formation in English without the Latin pretentiousness.
$endgroup$
– David K
23 hours ago





$begingroup$
For future reference, "vertice" is not a word. You can talk about each vertex of a triangle. Vertices is the plural of vertex. It's not the usual way of forming a plural in English, because it isn't English--it's Latin. If you were to speak of multiple vertexes, however, I wouldn't complain. That's a legitimate alternative plural formation in English without the Latin pretentiousness.
$endgroup$
– David K
23 hours ago











2 Answers
2






active

oldest

votes


















7












$begingroup$

We'll prove the case for $n=3$ and then generalize for an $n-$gon.





Let $triangle ABC$ be a triangle and $P$ an arbitraty point (inside or outside the triangle). Consider now the circumferences $omega$ and $tau$ with diameters $AP$ and $PB$ respectively. Consider furthermore the point $Din [AB]$, such that $PDbot AB$. In vitue of the converse of Thales' Theorem
$$angle ADP=90°implies Din omega qquadqquad angle PDB=90°implies Din tau$$ Analogously, we can prove that the intersections of the circles are $P$ and three points $D, Ein BC, Fin AC$ which lie on the sides of the triangles or on the extensions respectively.





Observe now that the triangles $triangle PBE, triangle PEC, triangle PCF, triangle PFA, triangle PDA$ and $triangle PBD$ are respectively inscribed in the circles $omega, tau$ and $rho$ (with the diameter $PC$). Thus, so is $triangle ABC$.




Now, once proven the case for a triangle, simply separate an $n-$gon into $(n-2)$ triangles, which will all be covered by the overlapping circles. Thus, the $n$-gon will also be covered $quadsquare$





For further reading: Episodes in nineteenth and twentieth century Euclidean Geometry (Honsberger), pages 79-86: Miquel's Theorem






share|cite|improve this answer











$endgroup$




















    7












    $begingroup$

    The property you found boils down to the following: given a segment $AB$ and any point $P$ outside it, then the circles having $PB$ and $PA$ as diameter completely cover triangle $ABP$. And that is obvious, because those circles are the circumcircles of triangles $APH$ and $BPH$, where $H$ is the orthogonal projection of $P$ onto line $AB$.



    enter image description here






    share|cite|improve this answer









    $endgroup$












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      2 Answers
      2






      active

      oldest

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      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      7












      $begingroup$

      We'll prove the case for $n=3$ and then generalize for an $n-$gon.





      Let $triangle ABC$ be a triangle and $P$ an arbitraty point (inside or outside the triangle). Consider now the circumferences $omega$ and $tau$ with diameters $AP$ and $PB$ respectively. Consider furthermore the point $Din [AB]$, such that $PDbot AB$. In vitue of the converse of Thales' Theorem
      $$angle ADP=90°implies Din omega qquadqquad angle PDB=90°implies Din tau$$ Analogously, we can prove that the intersections of the circles are $P$ and three points $D, Ein BC, Fin AC$ which lie on the sides of the triangles or on the extensions respectively.





      Observe now that the triangles $triangle PBE, triangle PEC, triangle PCF, triangle PFA, triangle PDA$ and $triangle PBD$ are respectively inscribed in the circles $omega, tau$ and $rho$ (with the diameter $PC$). Thus, so is $triangle ABC$.




      Now, once proven the case for a triangle, simply separate an $n-$gon into $(n-2)$ triangles, which will all be covered by the overlapping circles. Thus, the $n$-gon will also be covered $quadsquare$





      For further reading: Episodes in nineteenth and twentieth century Euclidean Geometry (Honsberger), pages 79-86: Miquel's Theorem






      share|cite|improve this answer











      $endgroup$

















        7












        $begingroup$

        We'll prove the case for $n=3$ and then generalize for an $n-$gon.





        Let $triangle ABC$ be a triangle and $P$ an arbitraty point (inside or outside the triangle). Consider now the circumferences $omega$ and $tau$ with diameters $AP$ and $PB$ respectively. Consider furthermore the point $Din [AB]$, such that $PDbot AB$. In vitue of the converse of Thales' Theorem
        $$angle ADP=90°implies Din omega qquadqquad angle PDB=90°implies Din tau$$ Analogously, we can prove that the intersections of the circles are $P$ and three points $D, Ein BC, Fin AC$ which lie on the sides of the triangles or on the extensions respectively.





        Observe now that the triangles $triangle PBE, triangle PEC, triangle PCF, triangle PFA, triangle PDA$ and $triangle PBD$ are respectively inscribed in the circles $omega, tau$ and $rho$ (with the diameter $PC$). Thus, so is $triangle ABC$.




        Now, once proven the case for a triangle, simply separate an $n-$gon into $(n-2)$ triangles, which will all be covered by the overlapping circles. Thus, the $n$-gon will also be covered $quadsquare$





        For further reading: Episodes in nineteenth and twentieth century Euclidean Geometry (Honsberger), pages 79-86: Miquel's Theorem






        share|cite|improve this answer











        $endgroup$















          7












          7








          7





          $begingroup$

          We'll prove the case for $n=3$ and then generalize for an $n-$gon.





          Let $triangle ABC$ be a triangle and $P$ an arbitraty point (inside or outside the triangle). Consider now the circumferences $omega$ and $tau$ with diameters $AP$ and $PB$ respectively. Consider furthermore the point $Din [AB]$, such that $PDbot AB$. In vitue of the converse of Thales' Theorem
          $$angle ADP=90°implies Din omega qquadqquad angle PDB=90°implies Din tau$$ Analogously, we can prove that the intersections of the circles are $P$ and three points $D, Ein BC, Fin AC$ which lie on the sides of the triangles or on the extensions respectively.





          Observe now that the triangles $triangle PBE, triangle PEC, triangle PCF, triangle PFA, triangle PDA$ and $triangle PBD$ are respectively inscribed in the circles $omega, tau$ and $rho$ (with the diameter $PC$). Thus, so is $triangle ABC$.




          Now, once proven the case for a triangle, simply separate an $n-$gon into $(n-2)$ triangles, which will all be covered by the overlapping circles. Thus, the $n$-gon will also be covered $quadsquare$





          For further reading: Episodes in nineteenth and twentieth century Euclidean Geometry (Honsberger), pages 79-86: Miquel's Theorem






          share|cite|improve this answer











          $endgroup$



          We'll prove the case for $n=3$ and then generalize for an $n-$gon.





          Let $triangle ABC$ be a triangle and $P$ an arbitraty point (inside or outside the triangle). Consider now the circumferences $omega$ and $tau$ with diameters $AP$ and $PB$ respectively. Consider furthermore the point $Din [AB]$, such that $PDbot AB$. In vitue of the converse of Thales' Theorem
          $$angle ADP=90°implies Din omega qquadqquad angle PDB=90°implies Din tau$$ Analogously, we can prove that the intersections of the circles are $P$ and three points $D, Ein BC, Fin AC$ which lie on the sides of the triangles or on the extensions respectively.





          Observe now that the triangles $triangle PBE, triangle PEC, triangle PCF, triangle PFA, triangle PDA$ and $triangle PBD$ are respectively inscribed in the circles $omega, tau$ and $rho$ (with the diameter $PC$). Thus, so is $triangle ABC$.




          Now, once proven the case for a triangle, simply separate an $n-$gon into $(n-2)$ triangles, which will all be covered by the overlapping circles. Thus, the $n$-gon will also be covered $quadsquare$





          For further reading: Episodes in nineteenth and twentieth century Euclidean Geometry (Honsberger), pages 79-86: Miquel's Theorem







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 19 hours ago

























          answered yesterday









          Dr. MathvaDr. Mathva

          2,857527




          2,857527





















              7












              $begingroup$

              The property you found boils down to the following: given a segment $AB$ and any point $P$ outside it, then the circles having $PB$ and $PA$ as diameter completely cover triangle $ABP$. And that is obvious, because those circles are the circumcircles of triangles $APH$ and $BPH$, where $H$ is the orthogonal projection of $P$ onto line $AB$.



              enter image description here






              share|cite|improve this answer









              $endgroup$

















                7












                $begingroup$

                The property you found boils down to the following: given a segment $AB$ and any point $P$ outside it, then the circles having $PB$ and $PA$ as diameter completely cover triangle $ABP$. And that is obvious, because those circles are the circumcircles of triangles $APH$ and $BPH$, where $H$ is the orthogonal projection of $P$ onto line $AB$.



                enter image description here






                share|cite|improve this answer









                $endgroup$















                  7












                  7








                  7





                  $begingroup$

                  The property you found boils down to the following: given a segment $AB$ and any point $P$ outside it, then the circles having $PB$ and $PA$ as diameter completely cover triangle $ABP$. And that is obvious, because those circles are the circumcircles of triangles $APH$ and $BPH$, where $H$ is the orthogonal projection of $P$ onto line $AB$.



                  enter image description here






                  share|cite|improve this answer









                  $endgroup$



                  The property you found boils down to the following: given a segment $AB$ and any point $P$ outside it, then the circles having $PB$ and $PA$ as diameter completely cover triangle $ABP$. And that is obvious, because those circles are the circumcircles of triangles $APH$ and $BPH$, where $H$ is the orthogonal projection of $P$ onto line $AB$.



                  enter image description here







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered yesterday









                  AretinoAretino

                  25.4k21445




                  25.4k21445



























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