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?

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. https://drive.google.com/file/d/0B8dFvepP2-ghRTJXRjBmMUVlSWc/view 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: https://drive.google.com/file/d/0B8dFvepP2-ghTnI2eHM5TzFIV0k/view 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