Unit IV: Chapter II, Parts A & B — Parallel Translation & Replacing¶
Julius Petersen — Methods and Theories (Authorized Translation by Sophus Haagensen, 1879 · pp. 46–60)
Chapter II — Transformation of the Figure · A. Parallel Translation & B. Replacing¶
[p. 46] In order to apply the locus methods of the First Chapter directly, the given elements of the sought figure must be adjoining to a certain extent, so that a substantial portion of the figure can be drawn at once and the remainder reduced to a single point or line. As soon as the given parts lie scattered, loci fail until we invoke our second great principle:
We must try of the drawn figure to form another, in which the given parts are thus gathered, that the construction may be carried out.
Once this transformed auxiliary figure has been constructed, reversing the transformation yields the original figure immediately. The three fundamental transformations of Chapter II are A. Parallel Translation, B. Replacing, and C. Revolution.
A. Parallel Translation (pp. 46–55)¶
[pp. 46–47] Parallel translation gathers dispersed sides, medians, diagonals, or angles by shifting certain segments to new positions parallel and equal to their original ones until their extremities meet in a constructible triangle or parallelogram:
Translation of the Triangle (Medians, p. 47): In \(\triangle ABC\) with medians meeting at centroid \(G\) (or \(O\) in Petersen’s diagram, shifting \(AB\) and \(AC\) to form parallelogram extensions), translating the sides or medians gathers a triangle whose three sides equal twice the three medians \((2m_a, 2m_b, 2m_c)\) and whose area is three times \(\triangle ABC\), solving Problem 246 (Construct a triangle, the three medians being given) at a single stroke, followed by Problems 247–258 combining altitudes and medians.
Translation of the Quadrilateral (pp. 48–49): In any quadrilateral \(ABCD\), translating \(AB\) and \(AD\) to \(CB_1\) and \(CD_1\) forms a parallelogram \(BD_1\) whose sides equal and parallel the diagonals \(AC, BD\), whose angles equal the angle between the diagonals, whose area is twice that of \(ABCD\), and whose own diagonals equal twice the segments joining the midpoints of opposite sides! Meanwhile the four sides of \(ABCD\) radiate from the single interior point \(C\).
Figure 4.1 — Petersen’s Problem 259 (p. 49): Trapezoid from its Four Sides via Parallel Translation. Translating leg BC (\(c = 13\)) parallel to upper base CD (\(b = 5\)) into ED gathers all four given lengths into the Euclidean three-side auxiliary triangle AED with base \(AE = a - b = 14\), \(AD = d = 15\), and \(ED = c = 13\). Reversing the translation along parallelogram \(EBCD\) yields ABCD on the exact integer lattice.
Canonical Parallel-Translation Paradigms (Problems 259–310)¶
Problem 259 (p. 49): Construct a trapezium (trapezoid), the four sides of which are given. Solution: Translate one of the sides, which are not parallel, to the extremity of the other, and we obtain a triangle, of which the three sides are known (\(a - b\), \(c\), and \(d\)). - Discussion: Solvable if and only if \(a \neq b\) and \(|d - c| < |a - b| < d + c\).
Placing a Segment Equal & Parallel to a Given Line Between Two Curves (pp. 51–52, Problems 271–274): To place a line with its extremities in two given curves \(K_1, K_2\), the line being equal and parallel to a given vector \(\vec{m}\). Translate curve \(K_1\) parallel to \(\vec{m}\) through distance \(|\vec{m}|\); its intersections with \(K_2\) are the exact endpoints! Applied in Problem 274 (p. 52) to marine navigation between two subtended-angle arcs.
Diminishing the Radius of a Circle to Naught (Circle Shrinking, pp. 52–53, Problems 275–278): [p. 52] A potent variant of parallel translation applies when a circle must touch given straight lines or circles: diminish the radius of the smallest given circle to zero so it shrinks to its centre point, while translating tangent lines parallel to themselves by that radius and expanding/contracting the other circles concentrically by \(\pm r\)! - Problem 275 (p. 52): To draw the common tangents to two circles \((O_1, R_1)\) and \((O_2, R_2)\) — reduced to drawing tangents from point \(O_2\) to concentric circles \((O_1, R_1 \mp R_2)\). - Problems 276–278 (pp. 52–53): Circle touching two lines and a circle, or touching a circle and a line at a given point.
B. Replacing (pp. 56–60, Problems 311–335)¶
[pp. 56–57] Replacing rearranges a sub-figure into a flipped or proportionally scaled congruent/similar position to achieve one of three ends:
To gather the given parts (Problem 311, p. 56): Inscribing a quadrilateral \(ABCD\) in a given circle given opposite sides \(AB, CD\) and the ratio \(BC : AD\) — flip \(\triangle ABC\) by interchanging \(A\) and \(C\) on the circle so the two known sides become adjacent!
To introduce given sums or differences of angles/sides (Problem 313, p. 57): Construct \(\triangle ABC\) from \(a, b\), and \(A - B\) by replacing \(\triangle ABC\) with \(A\) and \(B\) interchanged.
To make equal or proportional unknown lines cover each other (Problem 316, p. 57): Construct an inscribable (cyclic) quadrilateral \(ABCD\) whose four sides \(a, b, c, d\) are given — scale \(\triangle ABC\) by \(d : a\) and flip it into position \(ADC_1\); because \(\angle B + \angle D = 180^\circ\), \(DC\) and \(DC_1\) become collinear, eliminating the unknown circumcircle and yielding \(\triangle CAC_1\) directly via Locus f!