there is something wrong

本文由 nickname-ttp-818352 在 2015-08-07 發表於 "數學" 討論區

  1. nickname-ttp-818352

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    the mathematics test level 3 question 16 states that : let a,b,c be complex numbers that represent the two vertices of a triangle, then which of the following statements must be true?
    A:a+b+c is the orthocentre of the triangle
    B: a+b+c/3 is the centroid of the triangle
    C: a+b+c/2 is the centre of the nine-point circle of the triangle
    D: The product of abc is a real number
    the choice of answers are:
    a. A
    b. B
    c. C
    d. D
    But A,B and C are all correct. If there any typo of the question?
    #1 nickname-ttp-818352, 2015-08-07
  2. 56377379

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    本帖最後由 Mathematics tutor 1 於 2015-8-7 19:27 編輯

    Statement A is incorrect. Consider the trangle where a = 0, b = 1 and c = i, i.e.
    By definition, orthocente is the intersection of all altitudes, 0 is thus the orthocentre.
    However, a+b+c = 1+i is not the orthocentre.

    Statement C is incorrect. By definition, a nine-point circle is a circle that passes through the "nine" points:
    - the midpoints of all sides of the triangle;
    - the feet of all altitudes;
    - the midpoints of the line segments from each vertex of the triangle to the orthocentre.
    Using the same example above. Those "nine" points constitutes the four points as shown below:
    The centre of the nine-point circle of the triangle is thus the intersection of diagonals of the square with vertices 0, 1/2, (1/2)i, 1/2+(1/2)i, so the centre is 1/4+(1/4)i
    However, (a+b+c)/2 = (1+i)/2, which is not the centre.

    Statement D is also wrong. A counter-example will be the triangle with vertices a = -1, b = 1, and c = i, thus abc = -i, which is not a real number.

    It remains only Statement B is correct, which is indeed true.

    P.S. About your notation, one should write (a+b+c)/3 instead of a+b+c/3 to avoid confusion.

    --- by Andy
    #2 56377379, 2015-08-07