Congruent incircles are incircles of equal radius. One construction draws a cevian in a triangle that divides it into two triangles with equal inradii. The positions and sizes of these two incircles can then be determined by simultaneously solving the eight equations
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
for the eight variables 
, 
, 
, 
, 
, 
, 
, and 
, with 
, 
, and 
given. Generalizing to 
congruent circles gives the 
equations
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
for 
,
..., 
,
 |
(12)
|
for 
,
..., 
,
and
 |
(13)
|
to be solved for the unknowns 
and 
(
of them), 
and 
(
of each for 
, ..., 
), and 
, 
, 
, and 
, a total of 
unknowns.
For a single cevian 
dividing 
into two triangles with
equal inradii, write 
, 
, 
, 
, and let 
be the triangle area.
The cevian length 
, common inradius 
, and division of 
are given explicitly by
 |  |  |
(14)
|
 |  |  |
(15)
|
 |  |  |
(16)
|
 |  |  |
(17)
|
The length and radius formulas are given by Jackson and Takhaev (2015, p. 5). The equal-inradius condition gives 
, and Stewart's
theorem then gives 
. Yiu (2003) gives a systematic treatment of these
cevians.
Given an arbitrary triangle, let 
cevians be drawn from one of
its vertices so all of the 
triangles so determined have
equal incircles. The equal
incircles theorem says that the incircles determined
by spanning 2, 3, ..., 
adjacent triangles are also
equal (Wells 1991, p. 67).
