2 Typos corrected.

I have the vague idea that Hadamard is referring to the proof construction where you erect equilateral triangles BCA', CAB' and ABC' on the sides of the triangle, as described here. The Fermat point is the intersection of the cevians AA', BB' and CC'. It can also be constructed using the various angles of 60 resp. 90 120 degrees.

In the construction of the tangent from a point P to an ellipse with foci F and F' (in the book you cite), they consider an additional point f. The correspondence should be

F ↔ C,

F' ↔ A,

P ↔ B,

f ↔ P'.

The general philosophy behind both, I think, is to convert a sum of segments into a single segment.

1

I have the vague idea that Hadamard is referring to the proof where you erect equilateral triangles BCA', CAB' and ABC' on the sides of the triangle, as described here. The Fermat point is the intersection of the cevians AA', BB' and CC'. It can also be constructed using the various angles of 60 resp. 90 degrees.

In the construction of the tangent from a point P to an ellipse with foci F and F' (in the book you cite), they consider an additional point f. The correspondence should be

F ↔ C,

F' ↔ A,

P ↔ B,

f ↔ P'.

The general philosophy behind both, I think, is to convert a sum of segments into a single segment.