the

The

three of them. The center of each excircle is the intersection point of the inner bisector of the opposite angle together with the

two outer bisectors of the other angles. The relation of the bisectors to these circles is studied in Bisector1.html . Here is

studied another property of these circles, namely their definition as envelope of certain lines easily constructible from the

triangle. Following properties are valid:

1) Let {A',B',C'} be the projections of the incenter on the sides of ABC. Let also A

Then line AA' is the polar of A

2) Points {A',A

3) For any point P on line B'C' construct the intersection points {B'', C''} of line-pairs (PB,AC) and (PC,AB). Then line

B''C'' is tangent to the incircle.

4) The contact point D of the tangent B''C'' is the harmonic conjugate of the intersection point D' of B'C' and B''C'' with

respect to {B'', C''}.

To 1) The polar of A is B'C'. By reciprocity of polars the polar of A

the result.

To 2) From (1) follows that the pencil of lines at A: A(B,C,A',A

line it intercepts. In particular on BC.

To 3) This is a consequence of the Chasles-Steiner theorem on generating envelopes through lines defined in a similar way to

the present one. A discussion for this subject also in its present form is contained in section-8 of ThalesRemarks.html .

By applying this theorem, first we obtain the result that lines B''C'' envelope a conic to which are tangents the three

sides of the triangle {CA, AB, BC} respectively for the positions of P at {B', C', A

obtains also the positions of the lines which are symmetrics of sides {AC, AB} with respect to the bisectors at B and

C respectively. To see this it suffices to set P on B'C' respectively at its intersection with BI and CI. Thus we obtain

the result that the enveloping conic is tangent to three + two lines which are also tangent to the incircle. Thus the two

conics coincide.

To 4) This is proven in section-4 of Polar.html .

For example the excircle opposite to C can be generated as envelope of lines as shown in next figure (see also section-8 of

ThalesRemarks.html ).

In this figure {B',C'} are the projections of the excenter I

is joined to {B,C} to produce lines {PB, PC} which intersect the sides of the triangle respectively at {B'', C''}. In a way similar to t

he one of the previous section it can be seen that lines B''C'' envelope the excircle opposite to C with center I

Bisector1.html

ThalesRemarks.html

Polar.html

Produced with EucliDraw© |