From the definition follows that.

The circle is called the

Obviously.

3) y

This defines a quadratic equation for x.

x

The two roots x

Since the ellipse is also symmetric w.r to the axis AB it follows that.

This shows that.

The pair (F*, L*) has the same property as the (F, L) with respect to all points of the conic.

This constant is easily identified with |AB|=2ed/(1-e

Last quantity is called the

Using the definition again the ratio (y/y')

The form of this transformation implies that.

Also if SQR is a secant of the circle then it maps under f to a secant SQ'R' of the ellipse and this shows that.

In particular.

In fact, for any other point Q on SP and the projections Q', Q

This, by the orthogonality at Q

A consequence of the last property in combination with the symmetry of the ellipse is the fact.

The reason for this is the fact that quadrangles PFQP' and PF*P

Related to this property is also the other focal property of the ellipse by which.

In fact, define the reflexions S, T respectively of F, F* w.r. to the tangent PA and also the reflexion Q of F* with respect to the other tangent PB from P.

Triangles PTF and PQF are equal having three equal corresponding sides.

Thus, angles TPF and FPQ are equal, hence angle(FPF* + 2APF) = angle(FPF* + 2F*PB).

Which shows that angle(APF) = angle(F*PB).

A consequence of the last property is a couple of properties relating to the auxiliary circle.

The first property results by projecting F to P on the tangent at X.

Quadrangles FPST, FPRQ are then cyclic. By the previous property angle(TPF)=angle(QPR) and by the cyclic quadrangles this comes to angle(TSF)=angle(RFQ) thereby proving that angle(SFR) is a right one.

Also angle(TPQ) = angle(TPF)+angle(FPQ) = angle(TSF)+angle(FRQ) shows the other statement.

A consequence of this property is that.

This is seen by observing that the point of tangency X and the intersection point V of SR separate harmonically points (S,R). This in turn is seen by projecting X to point Y on the auxiliary with a parallel WX to the directrix. By circle properties (V,W) separate harmonically (T,Q), hence (V,X) separate harmonically (S,R). Thus the bundle of lines at F: F(V,S,X,R) is harmonic and two of its lines are orthogonal, hence they are bisectors of the other two (see Harmonic_Bundle.html ).

Properties 16), 17) imply two other ways to generate the ellipse.

Ellipse.html

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