Drawing a tangent to a circle at a given point in just 3 ruler-and-compass constructions

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A friend of mine recently introduced me to an interesting app called Euclidea that challenges you to complete geometric constructions like the ones we learnt in high school. The app has successfully made me feel quite silly about myself, for I am unable to solve this elementary problem:




Given a circle and a point on its circumference, construct the tangent to the circle at that point using ruler-and-compass constructions.




problem



The difficulty I am facing is in finding an optimal solution: I want to accomplish this using as few constructions as possible. So, drawing a line with the ruler would count as one construction, and drawing a circle with the pair of compasses would count as one construction.



I am able to do it in four constructions like this:



solution



But, apparently there is a solution using just three constructions. And, after several weeks of trying (and failing), I have decided to ask for help. Can anyone help me see the light?




euclidean-geometry geometric-construction




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    up vote
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    down vote

    favorite












    A friend of mine recently introduced me to an interesting app called Euclidea that challenges you to complete geometric constructions like the ones we learnt in high school. The app has successfully made me feel quite silly about myself, for I am unable to solve this elementary problem:




    Given a circle and a point on its circumference, construct the tangent to the circle at that point using ruler-and-compass constructions.




    problem



    The difficulty I am facing is in finding an optimal solution: I want to accomplish this using as few constructions as possible. So, drawing a line with the ruler would count as one construction, and drawing a circle with the pair of compasses would count as one construction.



    I am able to do it in four constructions like this:



    solution



    But, apparently there is a solution using just three constructions. And, after several weeks of trying (and failing), I have decided to ask for help. Can anyone help me see the light?




    euclidean-geometry geometric-construction




    share|cite|improve this question























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      A friend of mine recently introduced me to an interesting app called Euclidea that challenges you to complete geometric constructions like the ones we learnt in high school. The app has successfully made me feel quite silly about myself, for I am unable to solve this elementary problem:




      Given a circle and a point on its circumference, construct the tangent to the circle at that point using ruler-and-compass constructions.




      problem



      The difficulty I am facing is in finding an optimal solution: I want to accomplish this using as few constructions as possible. So, drawing a line with the ruler would count as one construction, and drawing a circle with the pair of compasses would count as one construction.



      I am able to do it in four constructions like this:



      solution



      But, apparently there is a solution using just three constructions. And, after several weeks of trying (and failing), I have decided to ask for help. Can anyone help me see the light?




      euclidean-geometry geometric-construction




      share|cite|improve this question













      A friend of mine recently introduced me to an interesting app called Euclidea that challenges you to complete geometric constructions like the ones we learnt in high school. The app has successfully made me feel quite silly about myself, for I am unable to solve this elementary problem:




      Given a circle and a point on its circumference, construct the tangent to the circle at that point using ruler-and-compass constructions.




      problem



      The difficulty I am facing is in finding an optimal solution: I want to accomplish this using as few constructions as possible. So, drawing a line with the ruler would count as one construction, and drawing a circle with the pair of compasses would count as one construction.



      I am able to do it in four constructions like this:



      solution



      But, apparently there is a solution using just three constructions. And, after several weeks of trying (and failing), I have decided to ask for help. Can anyone help me see the light?





      euclidean-geometry geometric-construction





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Jul 28 at 6:41
























      asked Jul 28 at 6:34









      Brahadeesh

      3,39831246




      3,39831246




















          1 Answer
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          1. Pick an arbitrary point $A$ on the circle, close to $P$, the point of tangency. (0 moves)


          2. Draw a circle centred at $A$ passing through $P$ and intersecting the circle again at $B$. (1 move)


          3. Draw a circle centred at $P$ passing through $B$. Let it intersect the circle centred at $A$ at $C$. (2 moves)


          4. Draw $PC$. (3 moves)






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          • Lovely! That was quick. :D
            – Brahadeesh
            Jul 28 at 6:46











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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          2
          down vote



          accepted










          1. Pick an arbitrary point $A$ on the circle, close to $P$, the point of tangency. (0 moves)


          2. Draw a circle centred at $A$ passing through $P$ and intersecting the circle again at $B$. (1 move)


          3. Draw a circle centred at $P$ passing through $B$. Let it intersect the circle centred at $A$ at $C$. (2 moves)


          4. Draw $PC$. (3 moves)






          share|cite|improve this answer





















          • Lovely! That was quick. :D
            – Brahadeesh
            Jul 28 at 6:46















          up vote
          2
          down vote



          accepted










          1. Pick an arbitrary point $A$ on the circle, close to $P$, the point of tangency. (0 moves)


          2. Draw a circle centred at $A$ passing through $P$ and intersecting the circle again at $B$. (1 move)


          3. Draw a circle centred at $P$ passing through $B$. Let it intersect the circle centred at $A$ at $C$. (2 moves)


          4. Draw $PC$. (3 moves)






          share|cite|improve this answer





















          • Lovely! That was quick. :D
            – Brahadeesh
            Jul 28 at 6:46













          up vote
          2
          down vote



          accepted







          up vote
          2
          down vote



          accepted






          1. Pick an arbitrary point $A$ on the circle, close to $P$, the point of tangency. (0 moves)


          2. Draw a circle centred at $A$ passing through $P$ and intersecting the circle again at $B$. (1 move)


          3. Draw a circle centred at $P$ passing through $B$. Let it intersect the circle centred at $A$ at $C$. (2 moves)


          4. Draw $PC$. (3 moves)






          share|cite|improve this answer













          1. Pick an arbitrary point $A$ on the circle, close to $P$, the point of tangency. (0 moves)


          2. Draw a circle centred at $A$ passing through $P$ and intersecting the circle again at $B$. (1 move)


          3. Draw a circle centred at $P$ passing through $B$. Let it intersect the circle centred at $A$ at $C$. (2 moves)


          4. Draw $PC$. (3 moves)







          share|cite|improve this answer













          share|cite|improve this answer



          share|cite|improve this answer











          answered Jul 28 at 6:43









          Sharky Kesa

          656312




          656312











          • Lovely! That was quick. :D
            – Brahadeesh
            Jul 28 at 6:46

















          • Lovely! That was quick. :D
            – Brahadeesh
            Jul 28 at 6:46
















          Lovely! That was quick. :D
          – Brahadeesh
          Jul 28 at 6:46





          Lovely! That was quick. :D
          – Brahadeesh
          Jul 28 at 6:46













           

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