Let vertices of an acute-angled triangle are `A(z_1),B(z_2),a n dC(z_3)dot` If the origin `O` is he orthocentre of the triangle, then prove that `z_1( z )_2+( z )_1z_2=_2( z )_3+( z )_2z_3=z_3( z )_1+( z )_(3)z_1dot`

Explanation

To prove that for the vertices of an acute-angled triangle \( A(z_1), B(z_2), C(z_3) \) with the origin \( O \) as the orthocenter, the following equations hold:

\[ z_1 \overline{z_2} + \overline{z_1} z_2 = z_2 \overline{z_3} + \overline{z_2} z_3 = z_3 \overline{z_1} + \overline{z_3} z_1 \]

### Step-by-step Solution:

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