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X(1), X(7), X(9), X(55), X(57), X(218), X(277), X(3174)

excenters

vertices of the intouch triangle

Geometric properties :

The symbolic substitution SS{a - > √a} transforms K176 = pK(X32, X4) into K1059.

K1059 meets the circumcircle (O) at A, B, C and three other (always real) points Q1, Q2, Q3 which are the vertices of a triangle we shall call the Gergonne triangle. See the related Nagel triangle in K692.

The centroid and Lemoine point of the Gergonne triangle are X(21151) and X(21149) respectively.

These points are the common points (apart X7 which is the orthocenter of the triangle) of the rectangular hyperbolas passing through {X7, X110, X2574, X2575, X3211}, {X7, X9, X100, X2550, X3174, X3243}, {X7, X513, X517, X672, X901} respectively.

Any pK(Ω, P) passing through Q1, Q2, Q3 must have :

• its pole Ω on psK(X32 x X57, X57, X6),

• its pivot P on psK(X57, X85, X4),

with the following pairings{Ω,P} : {6,7}, {31,169}, {603,57}, {1407,14256}, {Ω4,4}, {Ω9,9} where Ω4, Ω9 are unlisted in ETC.

Ω4 = a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^6-2 a^5 b+a^4 b^2-a^2 b^4+2 a b^5-b^6-2 a^5 c-2 a^4 b c+2 a^3 b^2 c+2 a^2 b^3 c+a^4 c^2+2 a^3 b c^2-2 a^2 b^2 c^2-2 a b^3 c^2+b^4 c^2+2 a^2 b c^3-2 a b^2 c^3-a^2 c^4+b^2 c^4+2 a c^5-c^6) : : , SEARCH = 0.0493801246097826, on the lines {X19, X614}, {X31, X607}, etc.

Ω9 = a^3 (a^2-2 a b+b^2-2 a c+c^2) : : , SEARCH =-0.141685370070000, on the lines {X6, X31}, {X19, X2195}, {X32, X1802}, {X40, X595}, {X44, X3059}, {X48, X2175}, {X57, X2191}, {X65, X1279}, {X109, X269}, etc.

Ω4, Ω9 are now X(21058), X(21059) in ETC (2018-08-17).