Does $\triangle ABC$ exist such that $\triangle ABC \sim \triangle DEF$, with $D, E, F$ being the incentre, centroid, orthocentre of $\triangle ABC$? Question:

Does $\triangle ABC$ exist such that $\triangle ABC \sim \triangle DEF$, with $D, E, F$ being the incentre, centroid, orthocentre of $\triangle ABC$, resp.?

For such a triangle to exist, it must be obtuse. Besides that, I have no idea how to prove or disprove it. For the case of $D, E, F$ being the orthocentre, centroid, circumcentre, it's impossible as they lie on the same line (Euler's line). That's the motivation of the problem. I have a feeling that brute force methods are needed (coordinate geometry). But I hate such an ugly approach. Any idea?

Yes, this triangle exists. (Found using brute force approximation.)

Coordinates:
$$A\approx(0.182,0.260)\quad B=(0,0)\quad C=(1,0)\\
D\approx(0.229,0.120)\quad E\approx(0.394,0.087)\quad F\approx(0.182,0.571)$$
Angles:
$$a=d\approx107.2957\quad b=e\approx55.0744\quad c=f\approx 17.6299$$
