Editing Incircle and excircles (section)

===Other excircle properties===
The circular [[convex hull|hull]] of the excircles is internally tangent to each of the excircles and is thus an [[Problem of Apollonius|Apollonius circle]].<ref>[http://forumgeom.fau.edu/FG2002volume2/FG200222.pdf Grinberg, Darij, and Yiu, Paul, "The Apollonius Circle as a Tucker Circle", ''Forum Geometricorum'' 2, 2002: pp. 175-182.]</ref> The radius of this Apollonius circle is <math>\tfrac{r^2 + s^2}{4r}</math> where <math>r</math> is the incircle radius and <math>s</math> is the semiperimeter of the triangle.<ref>[http://forumgeom.fau.edu/FG2003volume3/FG200320.pdf Stevanovi´c, Milorad R., "The Apollonius circle and related triangle centers", ''Forum Geometricorum'' 3, 2003, 187-195.]</ref>

The following relations hold among the inradius <math>r</math>, the circumradius <math>R</math>, the semiperimeter <math>s</math>, and the excircle radii <math>r_a</math>, <math>r_b</math>, <math>r_c</math>:<ref name=Bell>{{cite web |author=Bell, Amy |title="Hansen's right triangle theorem, its converse and a generalization", ''Forum Geometricorum'' 6, 2006, 335–342. |url=http://forumgeom.fau.edu/FG2006volume6/FG200639.pdf |access-date=2012-05-05 |url-status=dead |archive-url=https://web.archive.org/web/20210831080348/https://forumgeom.fau.edu/FG2006volume6/FG200639.pdf |archive-date=2021-08-31}}</ref>
:<math display=block>\begin{align}
              r_a + r_b + r_c &= 4R + r, \\
  r_a r_b + r_b r_c + r_c r_a &= s^2, \\
        r_a^2 + r_b^2 + r_c^2 &= \left(4R + r\right)^2 - 2s^2.
\end{align}</math>

The circle through the centers of the three excircles has radius <math>2R</math>.<ref name=Bell/>

If <math>H</math> is the [[orthocenter]] of <math>\triangle ABC</math>, then<ref name=Bell/>
:<math display=block>\begin{align}
          r_a + r_b + r_c + r &= \overline{AH} + \overline{BH} + \overline{CH} + 2R, \\
  r_a^2 + r_b^2 + r_c^2 + r^2 &= \overline{AH}^2 + \overline{BH}^2 + \overline{CH}^2 + (2R)^2.
\end{align}</math>