Incircle

1. Incircle

The incircle of a triangle ABC is the one which is tangent to all three sides of it and contained in the triangle. Its center is
the incenter of the triangle coinciding with the intersection point of the three inner bisectors (see Bisector0.html ).
The excircles of the triangle are the circles tangent to all three sides of the triangle and lying outside to it. There are
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 A₃ be the intersection of B'C' with BC.
    Then line AA' is the polar of A₃ with respect to the incircle.
2) Points {A',A₃} are harmonic conjugate with respect to {B,C}.
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 A₃ will pass through A. It passes also through A', hence
          the result.
To 2) From (1) follows that the pencil of lines at A: A(B,C,A',A₃) is harmonic. Hence it cuts a harmonic division on every
          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', A₃}. Then realize that line B''C''
          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 .

2. Excircles

There are analogous ways to generate also the excircles by considering a point on the line of contacts of the tangents from A.
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_c on sides {AC, AB} respectively. Then P moves on line B'C' and
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_c.

1. Incircle

2. Excircles

See Also