Let A,B and C represent the complex number `z_1, z_2, z_3` respectively on the complex plane. If the circumcentre of the triangle ABC lies on the origin, then the orthocentre is represented by the number

Options

A. `z_(1)+z_(2)-z_(3)`
B. `z_(2)+z_(3)-z_(1)`
C. `z_(3)+z_(1)-z_(2)`
D. `z_(1)+z_(2)+z_(3)`

Answer

To solve the problem step by step, we will analyze the relationship between the circumcenter, orthocenter, and centroid of triangle ABC represented by complex numbers \( z_1, z_2, z_3 \).

### Step 1: Understanding the Circumcenter and Orthocenter
The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect, and it is equidistant from all three vertices of the triangle. The orthocenter is the point where the altitudes of the triangle intersect.

### Step 2: Given Condition
We are given that the circumcenter of triangle ABC lies at the origin, which we denote as \( O = 0 \).

### Step 3: Relationship Between Centroid, Circumcenter, and Orthocenter
In any triangle, the relationship between the centroid \( G \), circumcenter \( O \), and orthocenter \( H \) is given by the formula:
\[ H = 2G - O \]
Since \( O = 0 \), this simplifies to:
\[ H = 2G \]

### Step 4: Finding the Centroid
The centroid \( G \) of triangle ABC formed by the complex numbers \( z_1, z_2, z_3 \) is given by:
\[ G = \frac{z_1 + z_2 + z_3}{3} \]

### Step 5: Calculating the Orthocenter
Substituting the expression for the centroid into the equation for the orthocenter:
\[ H = 2G = 2 \left( \frac{z_1 + z_2 + z_3}{3} \right) = \frac{2(z_1 + z_2 + z_3)}{3} \]

### Conclusion
Thus, the orthocenter \( H \) is represented by the complex number:
\[ H = \frac{2(z_1 + z_2 + z_3)}{3} \]

### Final Answer
The orthocenter is represented by the complex number \( \frac{2(z_1 + z_2 + z_3)}{3} \).

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