Theorem of Leibniz

The theorem of Leibniz in geometry is described as follows:

Maybe in the plan two fixed points has and B. Leibniz considers thus the place of the point M such as has ·(AM) ² + B·(BM) ² = cste. Either G the Barycentre of has (a) and B (b). Then the place, if there exists, is a circle of center G.

Demonstration

One develops ( AG + GM ) ² and of the same for BM = BG +GM .

The equality is reduced to (a+b) (GM) ² = cste which must be positive.

Remark : if + B has = 0, G is to some extent rejected ad infinitum (a physicist will consider that + B has = epsilon): the place is then an orthogonal line of the plan with AB.

The theorem spreads easily at several points has, B, C.

Relationship with the analysis situs

Leibniz in its geometrical characteristic represents the writing of the circle in the following way: ABC γ ABY which can be read " Similar ABC that ABY". In other words, being given three fixed points of space, B, and C, which form has describes the whole of the points Y which keep the same relation that a.c. with has and B? One can still translate this manner: AC γ AY and BC γ BY (the relation of C with has is the same one as of Y with has and the relation of C with B is the same one as of Y with B - equal distances).

See too

Barycentre

External bonds

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Theorem of Leibniz

Demonstration

Relationship with the analysis situs

See too

Related article

External bonds