Unit 5 Summary: Symmedian Geometry, Lemoine’s Point & Lemoine Circles¶
Mathematical Synthesis of Casey’s Supplementary Chapter, Sections I & III (Part 1)
1. Core Algebraic & Geometric Structure¶
Unit 5 opens Casey’s celebrated Supplementary Chapter on the Recent Elementary Geometry of the Triangle:
Isogonal Transformation: In homogeneous trilinear coordinates $(\alpha : \beta : \gamma)$, isogonal conjugation acts as componentwise inversion $(1/\alpha : 1/\beta : 1/\gamma)$; in barycentric coordinates $(u : v : w)$, it acts as $(a^2/u : b^2/v : c^2/w)$.
Lemoine’s Symmedian Point $K = X(6)$: Mapping the Centroid $G = (1 : 1 : 1)$ under isogonal conjugation yields $K = (a^2 : b^2 : c^2)$, whose trilinear distances $(x, y, z) = \frac{2\Delta}{a^2+b^2+c^2}(a, b, c)$ minimize $x^2 + y^2 + z^2$.
Lemoine’s Two Six-Point Circles: - First Lemoine Circle (`(Ml D1)`): Cut by the three parallels to the sides through $K$; centered at $M_L = X(182) = \operatorname{midpoint}(O, K)$ with side chords $\frac{a^3}{a^2+b^2+c^2}, \frac{b^3}{a^2+b^2+c^2}, \frac{c^3}{a^2+b^2+c^2}$ in triplicate ratio $a^3 : b^3 : c^3$. - Cosine (Second Lemoine) Circle (`(K Cb2)`): Cut by the three antiparallels to the sides through $K$; centered at $K$ itself with radius $R\tan\omega = \frac{abc}{a^2+b^2+c^2}$ and side chords proportional to $\cos A : \cos B : \cos C$.
2. Verified Exact Ledger on $\triangle ABC$ $(15, 13, 14)$ (geometor.casey.brocard)¶
Constructed in construct_symmedian_and_lemoine_circles() (0205.g1.svg):
Geometric Object |
Symbol |
Exact Symbolic Value |
|---|---|---|
Sum of Squares of Sides |
$a^2 + b^2 + c^2$ |
$15^2 + 13^2 + 14^2 = 590$ |
Lemoine / Symmedian Point |
$K = X(6)$ |
$\left(\frac{1673}{295}, \frac{1176}{295}\right)$ |
Minimum Squared-Distance Sum |
$d_a^2 + d_b^2 + d_c^2$ |
$\frac{4\Delta^2}{a^2+b^2+c^2} = \frac{14112}{295}$ |
First Lemoine Circle Centre |
$M_L = X(182)$ |
$\left(\frac{1869}{295}, \frac{19143}{4720}\right)$ |
First Lemoine Squared Radius |
$R_{L_1}^2$ |
$\frac{19477081}{891136}$ |
Triplicate-Ratio Chords |
$|D_1 D_2| : |E_1 E_2| : |F_1 F_2|$ |
$3375 : 2197 : 2744 = 15^3 : 13^3 : 14^3$ |
Cosine Circle Squared Radius |
$R_{\cos}^2 = R^2 \tan^2\omega$ |
$\left(\frac{273}{59}\right)^2 = \frac{74529}{3481}$ |