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(b + c - a) x (c^2y^2 - b^2z^2) = 0

X(1), X(8), X(40), X(56), X(84), X(2122), X(2123)

vertices of the extouch triangle

infinite points of K691

vertices of the Nagel triangle

M1, M2 on the line X4)X(9)

their isogonal conjugates N1, N2 on the line X(1)X(6)

See explanations in page K691.

K692 = pK(X6, X8) is tangent at A, B, C to the cevian lines of X(56) and the polar conic of X(56) is the circum-conic through X(8), X(56).

The isotomic transform of K692 is pK(X76, X3596).

The symbolic substitution SS{a - > √a} transforms K178 into K692.


K692 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 Nagel triangle. See the related Gergonne triangle in K1059.

The centroid and Lemoine point of the Nagel triangle are X(5657) and X(21150) respectively.

These points are the common points (apart X8 which is the orthocenter of the triangle) of the rectangular hyperbolas passing through {X3, X8, X513, X517, X901, X1149}, {X8, X110, X1201, X2574, X2575, X3157}, {X1, X8, X40, X100, X2550}, {X8, X514, X516, X927, X3160} respectively. Note that the first one is the Jerabek hyperbola of the Nagel triangle.

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

• its pole Ω on psK(X560, X1, X6),

• its pivot P on psK(X1, X75, X1),

with the following pairings{Ω,P} : {6,8}, {48,1}, {604,P604}, {Ω4,4}, where P604 and Ω4 are unlisted in ETC.

pK(X6, X8) is K692 and pK(X48, X1) is K1061.

P604 = a (a+b-c) (a-b+c) (a^4-b^4+2 a^2 b c-2 a b^2 c-2 a b c^2+2 b^2 c^2-c^4) : : , SEARCH = -0.708329462811358, on the lines {X1,X4}, {X3,X227}, {X8,X1943}, {X10,X1038}, {X12,X975}, {X40,X109}, {X46,X603}, {X56,X998}, {X57,X961}, {X65,X222}, {X77,X1441}, etc.

Ω4 = a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^5+a^4 b-a b^4-b^5+a^4 c-2 a^2 b^2 c+b^4 c-2 a^2 b c^2+2 a b^2 c^2-a c^4+b c^4-c^5) : : , SEARCH = 0.199517176364335, on the lines {X1,X2138}, {X6,X1854}, {X19,X614}, {X42,X3195}, etc.

P604, Ω4 are now X(21147), X(21148) in ETC (2018-08-18).