The altitude form the vertices A, B and C of the triangle ABC meet its circumcircle at D,E and F, respectively . The complex number representing the points D,E, and F are `z_(1),z_(2)` and `z_(3)`, respectively. If `(z_(3) -z_(1))//(z_(2) -z_(1))` is purely real, then show that triangle ABC is right-angled at A.

Explanation

To solve the problem, we need to show that if \(\frac{z_3 - z_1}{z_2 - z_1}\) is purely real, then triangle \(ABC\) is right-angled at \(A\). Here are the steps to arrive at the solution:

### Step 1: Understand the Geometry
We have triangle \(ABC\) with altitudes from vertices \(A\), \(B\), and \(C\) meeting the circumcircle at points \(D\), \(E\), and \(F\) respectively. The complex numbers representing these points are \(z_1\), \(z_2\), and \(z_3\).

### Step 2: Set Up the Condition
We are given that:
\[
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