Isodynamic & isogonic simplexes

Summary A simplex in an n-space and its tangential simplex formed of the n+1 tangent hyperplanes of its circumhypersphere (0) at its vertices are polar reciprocal of each other w. r. t. (0). The n+1 joins of their corresponding vertices, in general, do not concur 3, p.41, Ex.7;4;18;24;25]. But when n=2, the3 joins of the corresponding vertices of a triangle and its tangential triangle always concur at itsLemoine point L as its3 symmedians which are the isogonal conjugates of its medians w. r. it 7] such that its circumcircle coincides with the polar conic of L w. r. t. it 9]. For n=3, the4 joins of the corresponding vertices of a tetrahedron and its tangential tetrahedron concur at itsLemoine point L, if and only if it is isodynamic 5], as its4 Lemoinians 17] (called symmedians byCourt 6] which join its vertices to theLemoine points of its opposite faces (but not as the isogonal conjugates of its medians w. r. t. it) such that its circumsphere coincides with the polar quadric 16] of L w. r. t. it. The purpose of this paper is to develope analogonsly the theory of an isodynamic simplex, in an n-space (n>3), which is in rerspective with its tangential simplex, called isogonic under the circumstances. Their relationship as cevian & anticevian simplexes, and their association with S-configurations and related cevian quadrics are also pointed out. To Eurico Bompiani on his scientific Jubilee