Three lines moving at same velocity form a triangle. What is the formula for their lengths at time t?











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Three infinite lines on a plane forming a triangle where they cross. The lines are travelling perpendicular to their length at a constant speed $v=1$ such as the triangle is getting smaller.



The lengths of the sides of the triangle at time $t=0$ are $(x_0,y_0,z_0)$. Find the lengths of the sides of $t$.



This is a question I invented while thinking about physics. Extra marks for an intuitive solution without using vectors.



Considering the space-time diagram at some time $t$ all three lines will meet a point forming a triangular pyramid in space-time. Maybe this gives a clue to solving it.










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    up vote
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    Three infinite lines on a plane forming a triangle where they cross. The lines are travelling perpendicular to their length at a constant speed $v=1$ such as the triangle is getting smaller.



    The lengths of the sides of the triangle at time $t=0$ are $(x_0,y_0,z_0)$. Find the lengths of the sides of $t$.



    This is a question I invented while thinking about physics. Extra marks for an intuitive solution without using vectors.



    Considering the space-time diagram at some time $t$ all three lines will meet a point forming a triangular pyramid in space-time. Maybe this gives a clue to solving it.










    share|cite|improve this question
























      up vote
      5
      down vote

      favorite
      1









      up vote
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      down vote

      favorite
      1






      1





      Three infinite lines on a plane forming a triangle where they cross. The lines are travelling perpendicular to their length at a constant speed $v=1$ such as the triangle is getting smaller.



      The lengths of the sides of the triangle at time $t=0$ are $(x_0,y_0,z_0)$. Find the lengths of the sides of $t$.



      This is a question I invented while thinking about physics. Extra marks for an intuitive solution without using vectors.



      Considering the space-time diagram at some time $t$ all three lines will meet a point forming a triangular pyramid in space-time. Maybe this gives a clue to solving it.










      share|cite|improve this question













      Three infinite lines on a plane forming a triangle where they cross. The lines are travelling perpendicular to their length at a constant speed $v=1$ such as the triangle is getting smaller.



      The lengths of the sides of the triangle at time $t=0$ are $(x_0,y_0,z_0)$. Find the lengths of the sides of $t$.



      This is a question I invented while thinking about physics. Extra marks for an intuitive solution without using vectors.



      Considering the space-time diagram at some time $t$ all three lines will meet a point forming a triangular pyramid in space-time. Maybe this gives a clue to solving it.







      geometry dynamical-systems






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      asked Nov 18 at 2:41









      zooby

      946616




      946616






















          2 Answers
          2






          active

          oldest

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          up vote
          3
          down vote



          accepted










          Put the incentre at the origin, the result is just homothety $r(t)=r(0)-t$, for $t<r(0)=r$ (where $r(t)$ is the inradius at time $t$) and hence also on the side lengths $x(t)=x(0)(1-frac{t}{r(0)})$, etc.






          share|cite|improve this answer























          • Excellent! Answer very quick!
            – zooby
            Nov 18 at 3:09


















          up vote
          1
          down vote













          A little more definitive answer..........



          enter image description here



          $$x_t = x (1 - frac{2 Vcdot t}{(x+z-y)tan(frac{A}{2})})$$



          $$text{where} zge yge x text{and} (x+z-y)tan(frac{A}{2})ge Vcdot t$$



          Substitute $y$ and $y_t$, or $z$ and $z_t$ for $x$ and $x_t$ to get $y_t$ or $z_t$ in the above formula.






          share|cite|improve this answer























          • Yes, but you don't need $tan(A/2)$ because you can calculate that from the sides of the triangle. $L^2 = (x+y-z)(x-y+z)(z-x+y)/(x+y+z)$ give or take a factor of 2.
            – zooby
            Nov 18 at 15:10












          • Yes, that works.
            – Phil H
            Nov 18 at 18:06











          Your Answer





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          2 Answers
          2






          active

          oldest

          votes








          2 Answers
          2






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          3
          down vote



          accepted










          Put the incentre at the origin, the result is just homothety $r(t)=r(0)-t$, for $t<r(0)=r$ (where $r(t)$ is the inradius at time $t$) and hence also on the side lengths $x(t)=x(0)(1-frac{t}{r(0)})$, etc.






          share|cite|improve this answer























          • Excellent! Answer very quick!
            – zooby
            Nov 18 at 3:09















          up vote
          3
          down vote



          accepted










          Put the incentre at the origin, the result is just homothety $r(t)=r(0)-t$, for $t<r(0)=r$ (where $r(t)$ is the inradius at time $t$) and hence also on the side lengths $x(t)=x(0)(1-frac{t}{r(0)})$, etc.






          share|cite|improve this answer























          • Excellent! Answer very quick!
            – zooby
            Nov 18 at 3:09













          up vote
          3
          down vote



          accepted







          up vote
          3
          down vote



          accepted






          Put the incentre at the origin, the result is just homothety $r(t)=r(0)-t$, for $t<r(0)=r$ (where $r(t)$ is the inradius at time $t$) and hence also on the side lengths $x(t)=x(0)(1-frac{t}{r(0)})$, etc.






          share|cite|improve this answer














          Put the incentre at the origin, the result is just homothety $r(t)=r(0)-t$, for $t<r(0)=r$ (where $r(t)$ is the inradius at time $t$) and hence also on the side lengths $x(t)=x(0)(1-frac{t}{r(0)})$, etc.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Nov 18 at 13:15

























          answered Nov 18 at 3:02









          user10354138

          6,324623




          6,324623












          • Excellent! Answer very quick!
            – zooby
            Nov 18 at 3:09


















          • Excellent! Answer very quick!
            – zooby
            Nov 18 at 3:09
















          Excellent! Answer very quick!
          – zooby
          Nov 18 at 3:09




          Excellent! Answer very quick!
          – zooby
          Nov 18 at 3:09










          up vote
          1
          down vote













          A little more definitive answer..........



          enter image description here



          $$x_t = x (1 - frac{2 Vcdot t}{(x+z-y)tan(frac{A}{2})})$$



          $$text{where} zge yge x text{and} (x+z-y)tan(frac{A}{2})ge Vcdot t$$



          Substitute $y$ and $y_t$, or $z$ and $z_t$ for $x$ and $x_t$ to get $y_t$ or $z_t$ in the above formula.






          share|cite|improve this answer























          • Yes, but you don't need $tan(A/2)$ because you can calculate that from the sides of the triangle. $L^2 = (x+y-z)(x-y+z)(z-x+y)/(x+y+z)$ give or take a factor of 2.
            – zooby
            Nov 18 at 15:10












          • Yes, that works.
            – Phil H
            Nov 18 at 18:06















          up vote
          1
          down vote













          A little more definitive answer..........



          enter image description here



          $$x_t = x (1 - frac{2 Vcdot t}{(x+z-y)tan(frac{A}{2})})$$



          $$text{where} zge yge x text{and} (x+z-y)tan(frac{A}{2})ge Vcdot t$$



          Substitute $y$ and $y_t$, or $z$ and $z_t$ for $x$ and $x_t$ to get $y_t$ or $z_t$ in the above formula.






          share|cite|improve this answer























          • Yes, but you don't need $tan(A/2)$ because you can calculate that from the sides of the triangle. $L^2 = (x+y-z)(x-y+z)(z-x+y)/(x+y+z)$ give or take a factor of 2.
            – zooby
            Nov 18 at 15:10












          • Yes, that works.
            – Phil H
            Nov 18 at 18:06













          up vote
          1
          down vote










          up vote
          1
          down vote









          A little more definitive answer..........



          enter image description here



          $$x_t = x (1 - frac{2 Vcdot t}{(x+z-y)tan(frac{A}{2})})$$



          $$text{where} zge yge x text{and} (x+z-y)tan(frac{A}{2})ge Vcdot t$$



          Substitute $y$ and $y_t$, or $z$ and $z_t$ for $x$ and $x_t$ to get $y_t$ or $z_t$ in the above formula.






          share|cite|improve this answer














          A little more definitive answer..........



          enter image description here



          $$x_t = x (1 - frac{2 Vcdot t}{(x+z-y)tan(frac{A}{2})})$$



          $$text{where} zge yge x text{and} (x+z-y)tan(frac{A}{2})ge Vcdot t$$



          Substitute $y$ and $y_t$, or $z$ and $z_t$ for $x$ and $x_t$ to get $y_t$ or $z_t$ in the above formula.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Nov 18 at 14:03

























          answered Nov 18 at 5:29









          Phil H

          3,8782312




          3,8782312












          • Yes, but you don't need $tan(A/2)$ because you can calculate that from the sides of the triangle. $L^2 = (x+y-z)(x-y+z)(z-x+y)/(x+y+z)$ give or take a factor of 2.
            – zooby
            Nov 18 at 15:10












          • Yes, that works.
            – Phil H
            Nov 18 at 18:06


















          • Yes, but you don't need $tan(A/2)$ because you can calculate that from the sides of the triangle. $L^2 = (x+y-z)(x-y+z)(z-x+y)/(x+y+z)$ give or take a factor of 2.
            – zooby
            Nov 18 at 15:10












          • Yes, that works.
            – Phil H
            Nov 18 at 18:06
















          Yes, but you don't need $tan(A/2)$ because you can calculate that from the sides of the triangle. $L^2 = (x+y-z)(x-y+z)(z-x+y)/(x+y+z)$ give or take a factor of 2.
          – zooby
          Nov 18 at 15:10






          Yes, but you don't need $tan(A/2)$ because you can calculate that from the sides of the triangle. $L^2 = (x+y-z)(x-y+z)(z-x+y)/(x+y+z)$ give or take a factor of 2.
          – zooby
          Nov 18 at 15:10














          Yes, that works.
          – Phil H
          Nov 18 at 18:06




          Yes, that works.
          – Phil H
          Nov 18 at 18:06


















           

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