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X₂₈₅₅–X₂₈₈₂¶

Simson-Moses SM(P, U) Points & Conjugates

Simson-Moses circumcircle points SM(P, U) generated by isotomic conjugation of circumcircle pairs (X₂₈₅₅–X₂₈₆₈) and their isogonal conjugates on the line at infinity (X₂₈₆₉–X₂₈₈₂).

X₂₈₅₅–X₂₈₈₂ (Simson-Moses SM(P, U) Points & Conjugates) — 15 representative centers (from 28 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

Group Preamble — SM Points (Simson-Moses points)

Centers X₂₈₅₅ to X₂₈₆₈ are examples of Simson-Moses points, defined by changing “isogonal” to “isotomic” in the definition of Simson-Ribgy points (X₂₆₈₇ to X₂₇₇₀. That is, for points P and U on the circumcircle of triangle ABC, the point of intersection of the lines (P and isotomic conjugate of U) and (U and isotomic conjugate of P) defines the Simson-Moses point of P and U, denoted by SM(P,U). If P and U are given in barycentric coordinates \(p : q : r\) and \(u : v : w\), then

(rv - qw)/(qv - rw) : (pw - ru)/(rw - pu) : (qu - pv)/(pu - qv) (barycentrics)

If P and U are given in trilinear coordinates \(p : q : r\) and \(u : v : w\), then

\(b^{2}\)\(c^{2}(rv - qw)\)/(\(b^{2}\)qv - \(c^{2}\)rw) : \(c^{2}\)\(a^{2}(pw - ru)\)/(\(c^{2}\)rw - \(a^{2}\)pu) : \(a^{2}\)\(b^{2}(qu - pv)\)/(\(a^{2}\)pu - \(b^{2}\)qv) (trilinears)

On Oct. 23, 2004, Peter Moses described the point for which the designation SM(P,U) is here introduced, and he noted that SM(P,U), like SR(P,U), lies on the circumcircle.

In the following list, the appearance of I, J, K means that RS(X_i,X_j) = X(K):

74,99,477
98,99,842 98,110,2879 98,111,2880 98,112,2881
99,100,1290 99,101,2690 99,102,2695 99,103,2688 99,104,2687 99,105,2752 99,106,2758 99,107,1304 99,108,2766 99,109,2689 99,110,476 99,111,2770 99,112,935

The definitions of SR and SM points depend on two kinds of isoconjugates (i.e., isogonal and isotomic). To generalize, suppose

X =\(x : y : z\), P =\(p : q : r\), U =\(u : v : w\)

are points, none on a sideline of triangle ABC. Then

(X-isoconjugate of P) =\(yzqr : zxrp : xypq\) and (X-isoconjugate of U) =\(yzvw : zxwu : xyuv\),

Denote these points by P(X) and U(X), respectively. It is easy to show using determinants that PU(X) and UP(X) meet in the point. That point defines the X-Simson point of :g:`P <P>` and :g:`U <U*>`, denoted by S(X;P,U):

S(X;P,U) = yz(qw - rv)/(qvy - rwz) : zx(ru - pw)/(rwz - pux) : xy(pv - qu)/(pux - qvy).

Theorem: If P and U are on the circumcircle, then S(X;P,U) is on the circumcircle. A “computer proof” is easy: solve the equations a/p + b/q + c/r = 0 and a/u + b/v + c/w = 0 for r and w, substitute for these in a/r + b/s + c/t, where r, s, t are trilinears for S(X;P,U), and find that a/r + b/s + c/t = 0.

To conclude, SR(P,U) = S(X₁;P,U) and SM(P,U) = S(X₂;P,U).