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X(6), X(40), X(55), X(198), X(612), X(3158), X(3304), X(33587), X(33588), X(33589), X(33590), X(47299), X(47300)

A'B'C' : vertices of the incentral triangle

Q1Q2Q3 : vertices of the Thomson triangle

R1R2R3 : vertices of the tangential triangle of the Thomson triangle

other points below

Geometric properties :

(contributed by Peter Moses, 2019-07-17)

K1112 and K172 share 9 identified points namely X(6), X(55), X(198) and the vertices of two Thomson related triangles.

See Table 27, paragraph Lemoine point of T(P), and also K1109, K1110, K1111.

Other points on K1112 :

• K1112 meets the line at infinity at the same points as pK(X37, P) and pK(Ω, X1) where

P = a^3+2 a^2 b-2 a b^2-b^3+2 a^2 c-2 b^2 c-2 a c^2-2 b c^2-c^3 : : , on the lines {X8,X2891}, {X69,X674}, SEARCH = 5.68232861151321.

Ω = a^3 (a+2 b+2 c) : : , on the lines {X6,X595}, {X31,X32}, {X58,X2176}, SEARCH = 0.610543300390116.

• K1112 meets the circumcircle (O) at the vertices of the Thomson triangle and three other points which are the antipodes of the points (apart A, B, C) where pK(X1333, P') meets (O).

P' = a (a+b) (a+c) (3 a+b+c) (a^3+a^2 b-a b^2-b^3+a^2 c-2 a b c+b^2 c-a c^2+b c^2-c^3) : : , on the lines {X21,X3062}, {X28,X1385}, {X40,X1817}, SEARCH = -16.0342503633266.