tkz-euclide

Points

11 Definition of points by transformation¶

These transformations are:

• translation;

• homothety;

• orthogonal reflection or symmetry;

• central symmetry;

• orthogonal projection;

• rotation (degrees or radians);

• inversion with respect to a circle.

11.1 \tkzDefPointBy¶

The transformation is chosen with the options. There are two macros: \tkzDefPointBy transforms a single point, and \tkzDefPointsBy a list of points. By default, the image of \(A\) is \(A'\). For example, we write:

The result is in tkzPointResult.

11.1.1 translation¶

11.1.2 reflection (orthogonal symmetry)¶

11.1.3 homothety and projection¶

11.1.4 projection¶

11.1.5 symmetry¶

11.1.6 rotation¶

11.1.7 rotation in radian¶

11.1.8 rotation with nodes¶

11.1.9 inversion¶

Inversion is the process of transforming points to a corresponding set of points known as their inverse points. Two points \(P\) and \(P'\) are said to be inverses with respect to an inversion circle having inversion center \(O\) and inversion radius \(k\) if \(P'\) is the perpendicular foot of the altitude of \(OQP\), where \(Q\) is a point on the circle such that \(OQ\) is perpendicular to \(PQ\).
The quantity \(k^2\) is known as the circle power (Coxeter 1969, p. 81).

(https://mathworld.wolfram.com/Inversion.html)

Some propositions:

• The inverse of a circle (not through the center of inversion) is a circle.

• The inverse of a circle through the center of inversion is a line.

• The inverse of a line (not through the center of inversion) is a circle through the center of inversion.

• A circle orthogonal to the circle of inversion is its own inverse.

• A line through the center of inversion is its own inverse.

• Angles are preserved in inversion.

Explanation:

Directly: (center \(O\), power \(k^2={OA}^2=OP \times OP'\))

11.1.10 inversion of lines: example 1¶

11.1.11 inversion of lines: example 2¶

11.1.12 inversion of lines: example 3¶

11.1.13 inversion of circle and homothety¶

11.1.14 inversion of a triangle with respect to the incircle¶

11.1.15 inversion: orthogonal circle with inversion circle¶

The inversion circle itself, circles orthogonal to it, and lines through the inversion center are invariant under inversion. If a circle meets the circle of inversion, the points of intersection are invariant, so they also lie on the inverse circle: see \(I\) and \(J\) in the next figure.

For a more complex example, see Pappus 51.25.

11.1.16 inversion negative¶

It is an inversion followed by a symmetry with center \(O\).

11.2 Transformation of multiple points: \tkzDefPointsBy¶

A variant of the previous macro, to define several images. You give the names of the images as arguments, or leave the argument empty: the names of the images are then formed from the names of the original points.

The images are \(B'\) and \(C'\).

The images are \(D\) and \(E\).

The image is \(B'\).

11.2.1 translation of multiple points¶

11.2.2 symmetry of multiple points: an oval¶