The equal incircles theorem states that equal inradii in a row of triangles with a common vertex
propagate to every larger block of adjacent triangles.
More precisely, let 
be a point, and let 
, 
, ..., 
be points in order on a line
not containing 
. If the incircles
of the triangles 
have the same inradius 
for 
, then, for each 
with 
,
the triangles 
have equal inradii
for 
.
To see this, let 
be the common altitude from 
to the line, and let 
be the inradius of 
. A proof due to A. Drei uses the
identity 
for a triangle
with inradius 
, altitude 
,
and base angles 
and 
.
Multiplying this identity over the 
adjacent base segments cancels
the tangent factors belonging to supplementary
angles and gives
 |
The right side is independent of 
,
so 
is also independent of 
,
proving the theorem.
