# there is something wrong

1. ### nickname-ttp-818352 Expand Collapse

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
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
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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: