Problem of Napoleon

In Géométrie planes, the problem of Napoleon consists in building with the compass only the center of a given circle. One often allots this problem and his demonstration to Napoleon I {{er}}, but it is not sure that this demonstration is of him. Admittedly, it is known for its taste for mathematics and its training of artillerist enables him to control the wheels of them. However, in the same time, the Italian Lorenzo Mascheroni publishes his Géométrie of the compass work in which he precisely studies constructions with the compass alone.

Construction

That is to say the circle C which one wants to determine the center. That is to say a point has C.

A C1 circle centered in has meeting C out of B and B'.

Two C2 circles centered out of B and B' and passing by has meet at the point C.

A C3 circle centered on C and passing by has C1 meeting in D and Of

Two C4 circles centered in D and Of and passing by has meet in the center of C.

Note: it is necessary, so that construction is realizable, to take for the ray of the circle C1, a too large quantity neither, nor too small . More precisely, it is necessary that this ray lies between half and the double of the radius of the circle C

Demonstration

The principle of the demonstration is the possibility of building with the compass only the length B ² /a if the lengths has and B are known. The demonstration is based on the properties of the right-angled triangle. In the figure attached, triangle ABA' is right-angled out of B and H is the foot height resulting from B, one can thus write the following equality:

AH × AA' = AB ²

Thus

AH = \ frac {b^2} {2a}

and

AC = \ frac {b^2} {has}

In the preceding construction industry, one twice finds a configuration of this type:

the points has, B and B' is on the circle of center O and of ray R, distances AB, AB', BC and B' C are worth R thus $AC = \ frac {R^2} {R}$
the points has, D and Of are on the circle of center C and ray $\ frac {R^2} {R}$ , distances DA, Of has, DX, Of X are worth R thus $AX = \ frac {R^2} {R^2/r} = r$ .

Item X is well the center of the circle (C)

See too

Category: Geometrical construction

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