ADGEOM 1541

[Angel Montesdeoca]:

 

Let ABC be a triangle, (Ia), (Ib), (Ic) its excircles and (Ap) "the"
Apollonius circle (Kimberling, TCCT 1998, p. 102)
The circle (Ja) is tangent to (Ia) and externally to (Ib) and (Ic);
similarly for (Jb) and (Jc).

If (J) is the circle which touches (Ja),(Jb),(Jc) internally then (Ap)
and (J) are tangent in X(3030).
The center of (J) is:
J = (a(a^2(b+c) + a(b^2+b c+c^2) - 3b c(b+c)) :...:...),
with (6-9-13)-search number 166.4953881815177657693948105

The barycentric equation of the circle (J) is:

a^2 y z + b^2 z x + c^2 x y
- (x + y + z) ((a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c + 5 a^3 b c
- 7 a b^3 c + b^4 c - a^3 c^2 + 16 a b^2 c^2 + 3 b^3 c^2 - a^2 c^3 - 7 a
b c^3 + 3 b^2 c^3 + a c^4 + b c^4) x/(4 (a^2 b + a b^2 + a^2 c - 7 a b
c + b^2 c + a c^2 + b c^2)) +
(a^4 b - a^3 b^2 - a^2 b^3 + a b^4 +a^4 c - 7 a^3 b c + 5 a b^3 c +
b^4 c + 3 a^3 c^2 +16 a^2 b c^2 - b^3 c^2 + 3 a^2 c^3 - 7 a b c^3 - b^2
c^3 + a c^4 + b c^4) y/(4 (a^2 b + a b^2 + a^2 c - 7 a b c + b^2 c + a
c^2 + b c^2)) +
(a^4 b + 3 a^3 b^2 + 3 a^2 b^3 + a b^4 + a^4 c - 7 a^3 b c + 16 a^2
b^2 c - 7 a b^3 c + b^4 c - a^3 c^2 - b^3 c^2 - a^2 c^3 + 5 a b c^3 -
b^2 c^3 + a c^4 + b c^4) z/(4 (a^2 b + a b^2 + a^2 c - 7 a b c + b^2 c +
a c^2 + b c^2))) = 0.

Best regards,

Angel Montesdeoca