Editing Incircle and excircles (section)

==Euler's theorem==
[[Euler's theorem in geometry|Euler's theorem]] states that in a triangle:
:<math display=block>(R - r)^2 = d^2 + r^2,</math>

where <math>R</math> and <math>r</math> are the circumradius and inradius respectively, and <math>d</math> is the distance between the [[circumcenter]] and the incenter.

For excircles the equation is similar:
:<math display=block>\left(R + r_\text{ex}\right)^2 = d_\text{ex}^2 + r_\text{ex}^2,</math>

where <math>r_\text{ex}</math> is the radius of one of the excircles, and <math>d_\text{ex}</math> is the distance between the circumcenter and that excircle's center.<ref name=Nelson>Nelson, Roger, "Euler's triangle inequality via proof without words", ''Mathematics Magazine'' 81(1), February 2008, 58-61.</ref>{{sfn|Johnson|1929|p=187}}<ref>[http://forumgeom.fau.edu/FG2001volume1/FG200120.pdf Emelyanov, Lev, and Emelyanova, Tatiana. "Euler's formula and Poncelet's porism", ''Forum Geometricorum'' 1, 2001: pp. 137–140.]</ref>