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Math Help - square with 3 Vertexes

  1. #1
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    square with 3 Vertexes

    If we know the three Vertexes of square ... Can we find the fourth Vertex?
    what are the equations of theses vertexes?
    Sides of the square is not parallel to axes Coordinateness.
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    Re: square with 3 Vertexes

    Quote Originally Posted by rqeeb View Post
    If we know the three Vertexes of square ... Can we find the fourth Vertex?
    what are the equations of theses vertexes?
    Sides of the square is not parallel to axes Coordinateness.
    1. Let A, B, C denote the known vertices.

    2. The point D is the intersection-point of the parallels to \overline{AB} and \overline{BC}. This case is drawn in blue.

    3. The point D is the image of B by reflection over \overline{AC}. This case is drawn in red.
    Attached Thumbnails Attached Thumbnails square with 3 Vertexes-vierteckpktquadrat.png  
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    Re: square with 3 Vertexes

    what about this solution
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    Re: square with 3 Vertexes

    Quote Originally Posted by rqeeb View Post
    what about this solution
    That looks fine to me.

    I've found a 3rd way to get the 4th vertex:

    Let A(A_x, A_y),\ B(B_x, B_y),\ C(C_x, C_y) denote the known vertices of the square.

    The midpoint of the square is M. Then the M has the coordinates:

    M\left(\dfrac{A_x+C_x}2,\ \dfrac{A_y+C_y}2 \right)

    M is simultaneously the midpoint of BD:

    \left(\dfrac{A_x+C_x}2,\ \dfrac{A_y+C_y}2 \right) = \left(\dfrac{B_x+D_x}2,\ \dfrac{B_y+D_y}2 \right)

    Calculate D_x, D_y!

    I've got: D\left(A_x+C_x-B_x,\ A_y+C_y-B_y \right)

    In my opinion this way seems to be the most simple one.
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    Re: square with 3 Vertexes

    Great work
    Thanks alot
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  6. #6
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    Re: square with 3 Vertexes

    Hello, rqeeb!

    If you make a sketch, the answer is obvious.


    If we know the three vertexes of square, can we find the fourth vertex?

    Suppose the three given vertices are: . A(1,5),\;B(2,2),\;C(5,3)

    Plot points A,B,C.

    Code:
          |               D
          |               ♠
          | (1,5)         : 1
          | A ♥ - - - - - .
          |         3
          |
          |
          |
          |                   C
          |                   ♥ (5,3)
          |                   :1
          |     B ♥ - - - - - .
          |     (2,2)   3
          |
      - - + - - - - - - - - - - - -
          |

    Going from B to C, we move right 3 and up 1.

    Since AD \parallel BC and AD = BC,
    . . going from A to D, we again move right 3 and up 1.

    Therefore, D is located at (4,6).

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  7. #7
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    Re: square with 3 Vertexes

    Quote Originally Posted by Soroban View Post
    Hello, rqeeb!

    If you make a sketch, the answer is obvious.



    Suppose the three given vertices are: . A(1,5),\;B(2,2),\;C(5,3)

    Plot points A,B,C.

    Code:
          |               D
          |               ♠
          | (1,5)         : 1
          | A ♥ - - - - - .
          |         3
          |
          |
          |
          |                   C
          |                   ♥ (5,3)
          |                   :1
          |     B ♥ - - - - - .
          |     (2,2)   3
          |
      - - + - - - - - - - - - - - -
          |

    Going from B to C, we move right 3 and up 1.

    Since AD \parallel BC and AD = BC,
    . . going from A to D, we again move right 3 and up 1.

    Therefore, D is located at (4,6).

    in case that the square sides parallel the axies. but if not???
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  8. #8
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    Re: square with 3 Vertexes

    Hello, rqeeb!

    in case that the square sides parallel the axes. . What?
    But if not?

    Did you even look at my graph?

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  9. #9
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    Re: square with 3 Vertexes

    Quote Originally Posted by rqeeb View Post
    in case that the square sides parallel the axies. but if not???
    Have a second look:
    Attached Thumbnails Attached Thumbnails square with 3 Vertexes-quad_soroban.png  
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