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Given circle c with center A and a chord HI, and a point G within segment HI, construct circle d with center C such that the radical axis passes through G and point F lies on both d, line HI and line CA.

TL;DR Given the things in blue, construct the things in red.

Intuition tells me that there should be an unique solution given the level of constraint. I have also constructed one point J on the red circle.

A purely geometric construction and proof, not utilizing analytic geometry, would be appreciated.

Thanks in advance. Given the things in blue, construct the things in red.

Ditto, with point J.

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1 Answer 1

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Once you constructed point $J$, you can construct $F$ as the intersection of the line $HI$ and the line perpendicular to $AJ$ passing through $J$. This is because $AF$ is a diameter of $d$. The rest of the construction is easy and I leave it to you.

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