The easiest way to prove this is to express the cross ratio (Q

Hence if this cross ratio is -1 for a line through P it is also -1 for all lines through P. Line QP' is called also the

f

In fact, this is a line through the intersection point Q of {L

sf

To find it notice that line PQ belongs also to that pencil, hence satisfies

sf

and consequently

(s,t) = k(-f

Thus, line PQ has the equation

-f

The result follows by applying the previous relation of the sines and the fact that the ratio of the distances of P from the two lines is proportional to the ratio of sines:

R

The symbol refers to the above figure and defines the affine reflection (see Affine_Reflexion_Basics.html ) with axes of fixed points the line PQ and

f(X) = (X,a) + c = 0,

where (X,Y)=x

Here JX denotes the π/2-rotated of vector X. Thus if X=(x

(a

and the conjugate direction

(a

Hence

a

a

From these follows that

(a

and finally

Affine_Reflexion_Basics.html

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