The Apollonius triangle of a reference triangle 
is the triangle
formed by the points of tangency
of its three excircles with their enclosing Apollonius
circle (Lozada 2025). Label its vertices 
, 
, and 
according to the excircles
opposite 
,
,
and 
,
respectively.
The figure shows the reference triangle in gray, its three excircles in orange, and the Apollonius
triangle in blue. The dashed radii join the center 
of the enclosing Apollonius
circle to its points of tangency
through the corresponding excenters 
, 
, and 
.
The Apollonius triangle is a type 1 central triangle. Its trilinear vertex matrix, with rows
corresponding to 
,
,
and 
,
is
![[-(a[a(b+c)+b^2+c^2]^2)/(2s) b(a+c)^2(a+b-c) c(a+b)^2(a-b+c); a(b+c)^2(a+b-c) -(b[b(c+a)+c^2+a^2]^2)/(2s) c(a+b)^2(-a+b+c); a(b+c)^2(a-b+c) b(a+c)^2(-a+b+c) -(c[c(a+b)+a^2+b^2]^2)/(2s)],](/images/equations/ApolloniusTriangle/NumberedEquation1.svg) |
where 
,
,
,
and 
is the semiperimeter of the reference
triangle.
The lines 
, 
, and 
concur at the Apollonius
point.
