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.
