Find certain points on circle when viewing from different point

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I have a circle like this:

1

Where there is a known line passing through the centre of the circle.

I want to find the points D and E on the image (left and right most part of the circle) when the circle is viewed from a different point, like this:

2

Obviously points D and E are not the left and right most part of the circle (or rather, ellipse from this pov). Is there any way I can draw a line through the centre of the circle/ellipse that passes points D and E? I know a perpendicular line of line B-C passing through the centre (point A) will not work.

Is there maybe any relation between the angle of line B-C and the angle of the line that can be drawn between points D and E?