Triangle given the incenter I, the centroid G and the orthocenter H

1) Take N collinear with H and G such that HN = 3NG (N is the ninecenter).
2) Take O collinear with H and G such that GO = 2NG (O is the circumcenter).
3) The circle through N tangent to IO at O cuts IN at N and M, IM is the diameter 2R of the circumcircle.
4) Let O' be the reflection of O on I. The length O'H is R-2r, r being the inradius.
5) Draw the MacBeath ellipse with focii O, H and major axis R.
6) The common tangents to the incircle and the MacBeath ellipse intersect on the circumcircle at three points that are the vertices of the requested triangle.
The problem is solvable if I is internal to the orthocentroidal circle.