Cevian Projections of Inscribed Triangles and Generalized Wallace Lines

Let Δ = ABC be a reference triangle and Δ′ = A′B′C′ an inscribed triangle of Δ. We define the cevian projection of Δ′ as the cevian triangle ΔP of a certain point P . Given a point P not on a sideline, all inscribed triangles with common cevian projection ΔP form a family DP = {Δ(t) = AtBtCt, t ∈ R}. The parallels of the lines AAt, BBt, CCt through any point of a certain circumconic CP intersect the sidelines a, b, c in collinear points X , Y , Z, respectively. This is a generalization of the well-known theorem of Wallace. 1. Notations Let Δ = ABC be a positive oriented reference triangle with the sidelines a, b, c. A point P in the plane of Δ is described by its homogeneous barycentric coordinates u, v, w in reference to Δ: P = (u : v : w), a line : ux+vy+wz = 0 by = [u : v : w]. For a point P = (u : v : w) not on a sideline, denote by ΔP = PaPbPc its cevian triangle with the vertices Pa = (0 : v : w), Pb = (u : 0 : w), Pc = (u : v : 0), (1) and the sidelines pa = [−vw : wu : uv], pb = [vw : −wu : uv], pc = [vw : wu : −uv]. (2) The directions of these sidelines (as points of intersection with the infinite line) are La = (u(v − w) : −v(w + u) : w(u+ v)) Lb = (u(v + w) : v(w − u) : −w(u+ v)) (3) Lc = (−u(v + w) : v(w + u) : w(u− v)) . The medial operator m on points maps P to the point mP = (v + w : w + u : u+ v) =: M (4) so that the centroid G divides the segment PM in the ratio 2 : 1 (see, for example, [4]). Publication Date: June 2, 2016. Communicating Editor: Paul Yiu. Thanks are due to Paul Yiu for his lively interest in the paper, for valuable suggestions and especially for the examples in section 3.