Unit II: Chapter I, Part A — Loci of Points & The Method of Two Loci — Summary¶
Structural & Heuristic Overview — Kapitel I.A — Geometriske Steder for Punkter (a–k) (pp. 7–22, 37–40 · Problems 1–116 & 202–219)
Architectural Overview (pp. 7–22, 37–40)¶
Chapter I, Part A builds the complete dictionary of elementary point loci (a–k) so that whenever a problem reduces to locating a single unknown point \(P\) subject to two conditions \(\mathcal{C}_1 \wedge \mathcal{C}_2\), omitting \(\mathcal{C}_2\) and \(\mathcal{C}_1\) in turn produces two straight-line or circular loci \(\mathcal{L}_1, \mathcal{L}_2\) whose intersections \(\mathcal{L}_1 \cap \mathcal{L}_2\) solve the problem directly.
Complete Dictionary of Point Loci (a – k)¶
Locus |
Detached Condition on Point \(P\) |
Resulting Constructible Curve |
|---|---|---|
a |
Given distance \(r\) from fixed point \(O\) (or tangent length / visual angle to a circle) |
Circle \((O, r)\) (or concentric circles \(O, R \pm r\)) |
b |
Given distance \(h\) from fixed straight line \(l\) (or fixed triangle area on base \(AB\)) |
Pair of straight lines parallel to \(l\) at distance \(\pm h\) |
c |
Equal distances \(PA = PB\) from two fixed points |
Perpendicular bisector of segment \(AB\) |
d |
Equal perpendicular distances from two fixed lines \(l_1, l_2\) |
Pair of orthogonal angle bisectors of \((l_1, l_2)\) |
e |
Segment \(AB\) seen under given angle \(v\) (Pothenot’s Problem 19) |
Pair of symmetric circular arcs on chord \(AB\) (single circle on diameter \(AB\) when \(v = 90^\circ\)) |
f |
Given distance ratio \(PA : PB = m : n\) (\(m \neq n\)) |
Circle of Apollonius on harmonic diameter \(CC_1\) where \((A, B; C, C_1) = -1\) |
g |
Given ratio \(m : n\) of distances from two intersecting lines |
Pair of lines through \(l_1 \cap l_2\) forming harmonical rays |
h |
Constant difference of squares \(PB^2 - PA^2 = a^2\) (or equal tangent powers to two circles) |
Straight line \(\perp AB\) (radical axis, concurring at radical centre in Problem 30) |
i |
Constant sum of squares \(PA^2 + PB^2 = a^2\) |
Circle centered at midpoint \(M\) of \(AB\) with \(r^2 = \tfrac{1}{2}a^2 - \tfrac{1}{4}AB^2\) |
k |
Constant sum or difference \(a\) of distances from two intersecting lines |
Four segments of a rectangle (sum) plus eight infinite prolongations (difference) |
Gen. |
Rigid sub-figure moved with one constraint omitted (pp. 37–40) |
Parallel lines or concentric circles traced by free vertices (Problems 202–219) |