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X₂₆₇₇–X₂₇₇₀¶

Rigby-Simson RS(P, U) & Simson-Rigby SR(P, U) Points

Rigby-Simson orthopoles RS(P, U) where the Simson lines of circumcircle points P and U meet (X₂₆₇₇–X₂₆₈₆), and Simson-Rigby circumcircle points SR(P, U) whose Simson lines pass through RS(P, U) perpendicular to PU (X₂₆₈₇–X₂₇₇₀).

X₂₆₇₇–X₂₇₇₀ (Rigby-Simson RS(P, U) & Simson-Rigby SR(P, U) Points) — 16 representative centers (from 94 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

Group Preamble — RS Points and SR Points (Rigby-Simson points and Simson-Rigby points)

Centers X₂₆₇₇ to X₂₇₇₀ are examples of Rigby-Simson points and Simson-Rigby points. These two types of points are discussed in

Ross Honsberger , Episodes in Nineteenth and Twentieth Century Euclidean Geometry , Mathematical Association of America, 1995. Chapter 11: The Orthopole, pages 124-136.

Specifically, on pages 132-136, Honsberger defines the “Rigby Point,” in honor of John Rigby, as the orthopole of the six sides of two triangles, all of whose vertices lie on a circle, and he notes that this orthopole lies on the six Simson lines of the vertices.

Now suppose that P =\(p : q : r\) (trilinears) and U =\(u : v : w\) are distinct points on the circumcircle of a triangle ABC. Their Simson lines meet in a point X which is here called the Rigby-Simson point of P and U, denoted by RS(P,U). That is, RS(P,U) is the orthopole of line PU with respect to triangle ABC.

On the circumcircle, there is a unique point K whose Simson line is perpendicular to line PU, and whose Simson line also passes through RS(P,U). The point K is here called the Simson-Rigby point of P and U, denoted by SR(P,U). Note that SR(P,U) = Λ(P,U)

If you have The Geometer’s Sketchpad, you can view RS(P,U) and SR(P,U) (RigbySimson.gsp).

Continuing with contributions of Peter Moses during October, 2004, trilinears for RS(P,U) are

\(f(a,b,c,p,q,r,u,v,w) : f(b,c,a,q,r,p,v,w,u) : f(c,a,b,r,p,q,w,u,v)\), where

\(f(a,b,c,p,q,r,u,v,w) = 1/[a^{2}r^{2}v^{2}(q \cos A - p \cos B)(w \cos A - u \cos C)] - 1/[a^{2}q^{2}w^{2}(r \cos A - p \cos C)(v \cos A - u \cos B)]\),

and trilinears for SR(P,U), reckoned as the point of intersection of lines P-to-\(U ^{- 1}\) and U-to-\(P ^{- 1}\), are

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

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

74,98,691 74,99,842 74,110,477 74,1113,125 74,1114,125
98,1379,115 98,1380,115
99,110,691
100,101,1308 100,104,953 100,110,1290
101,109,1521 101,929,1521
104,1381 104,1382
110,112,1554 110,476,1553 110,935,1554
112,935,1554
1113,1114,125
1379,1380,115
1381,1382,11

In the next list (from P. Moses, Oct. 15, 2004), the appearance of I, J, K means that SR(X_i,X_j) = X(K):

74,98,691 74,99,842 74,104,1290 74,110,477 74,841,1302 74,935,1297 74,1113,1114 74,1141,1291 74,1294,1304
98,110,842 98,843,1296 98,1379,1380
99,110,691 99,111,843 99,1379,1379 & 99,1380,1380 99,2378,2379
100,101,1308 100,104,953 100,105,840 100,109,2222 100,110,1290 100,1381,1381 100,1382,1382
101,109,929
102,103,929 102,104,2222 103,104,1308
104,840,1292 104,1381,1382
107,110,1304
110,112,935 110,827,1287 110,930,1291 110,1113,1113 110,1114,1114