Complex Number geometry

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In the Argand diagram shown below the complex numbers – 1 + i, 1 + i, 1 – i, – 1 – i represent the vertices of a square ABCD.
The equation of its diagonal BD is y = x. The complex number k + ki where – 1 < k < 0 represents the point E which is in the fourth quadrant and lies on the line y = x. EFGC is a square such that F lies on AB. The line GE meets the line CD
produced at H such that H is represented by the complex number – 2 – i.
Screen Shot 2021-12-03 at 10.47.51 pm.png


How do I show F = -k\(k+2) + i? I've tried rotating EC and GC by i, but none seem to get the answer, nor even a fraction with k
 

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In the Argand diagram shown below the complex numbers – 1 + i, 1 + i, 1 – i, – 1 – i represent the vertices of a square ABCD.
The equation of its diagonal BD is y = x. The complex number k + ki where – 1 < k < 0 represents the point E which is in the fourth quadrant and lies on the line y = x. EFGC is a square such that F lies on AB. The line GE meets the line CD
produced at H such that H is represented by the complex number – 2 – i. View attachment 1229

How do I show F = -k\(k+2) + i? I've tried rotating EC and GC by i, but none seem to get the answer, nor even a fraction with k

What course are you taking? I have never seen geometry at this level.
 

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