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Math Help - How to find triangles third vertex(x,y,z) if other two are known

  1. #1
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    Exclamation How to find triangles third vertex(x,y,z) if other two are known

    How to find triangles third vertex C(x,y,z) if we known:
    1. other two vertices A(x,y,z), B(x,y,z)
    2. lengths of all triangles sides
    3. the triangle's normal

    I understand how to solve it in a graph, but can't find an analytical solution.
    Last edited by kolkir; June 20th 2011 at 11:19 PM.
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  2. #2
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    Re: How to find triangles third vertex(x,y,z) if other two are known

    Quote Originally Posted by kolkir View Post
    How to find triangles third vertex C(x,y,z) if we known:
    1. other two vertices A(x,y,z), B(x,y,z)
    2. lengths of all triangles sides
    3. the triangles normal
    I have no idea what is meant by #3. Normal?

    If #2 means that we know the lengths, \overline{AC}~\&~\overline{BC} then the point C is either of the intersections of the circle centered at A with radius \overline{AC} and the circle centered at B with radius \overline{BC}.
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  3. #3
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    Re: How to find triangles third vertex(x,y,z) if other two are known

    Thanks, for the answer. Yes #3 is normal or the triangle. We can't use solution with circle because i need three coordinates for C point. I have found one solution, but it is not very elegant:
    1. calculate angle between (AC and AB) and (BA and BC) in 2D - where we can easily calculate C coordinates (by the low of cosines)
    2. rotate AB vector about normal with two angles - which give us two vectors(lines)
    3. find point of intersection on these two lines - it will be our C point.
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  4. #4
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    Re: How to find triangles third vertex(x,y,z) if other two are known

    You did not understand Plato's question. What do you mean by "the normal to a triangle". Do you mean the normal to the plane determined by the triangle? I think Plato was assuming you were working in a given plane. Of course, being given the normal to the plane means you are given a plane.

    But what I would do is this: suppose we are given points A(a_1, a_2, a_3) and B(b_1, b_2, b_3) as well as normal vector <X, Y, Z> and are told that the length of AC is r_a, the length of BC is r_b. Let C be the unknown point (c_1, c_2, c_3).

    Then we seek the intersection of three surfaces: the sphere (x-a_1)^2+ (y- a_2)^2+ (z- a_3)^2= r_a^2, the sphere (x- b_1)^2+ (y- b_2)^2+ (z- b_3)^2= r_b^2, and the plane X(x- a_1)+ Y(y- a_2)+ Z(z- a_3)= 0. That gives three equations to solve for x, y, and z.
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  5. #5
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    Re: How to find triangles third vertex(x,y,z) if other two are known

    Yes, saying "the normal to a triangle" i meant "the normal to the plane determined by the triangle".
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