From Equilateral to 30–60–90: An Invariant Family of the Orthic Map

The orthic (pedal) map sends an acute triangle to the triangle formed by the feet of its altitudes, acting on angles by A ↦ 180° − 2A. This map has exactly one fixed angle, 60°, so the triangles carrying a 60° angle form an invariant one-parameter family T(d) = (60°, 60° − d, 60° + d). We follow that family across its entire acute range: at d = 0 it is the equilateral triangle, at d = 30° it is the 30–60–90 triangle, and between them the map is a linear doubling while the contraction per step falls from 1/4 to 0. Everything here is classical; the contribution is the isolation of this family and an exact rendering of the passage. The deposit includes an animation sweeping d from 0° to 30°, drawn from exact altitude-foot constructions.