Unit 6 Summary: Brocard Geometry, Tucker/Taylor Circles & Associated Triangle Points¶
Mathematical Synthesis of Casey’s Supplementary Chapter, Sections II–VII & Misc. Exercises
1. Unified Brocard–Steiner–Tarry Architecture¶
Unit 6 completes the Supplementary Chapter by uniting spiral similarity, isogonal bicentric pairs, and circumcircle concurrencies:
Brocard Angle $\omega$: Governed purely by rational quadratic invariants via $\cot\omega = \frac{a^2 + b^2 + c^2}{4\Delta}$ and Heron’s identity $(a^2+b^2+c^2)^2 + 16\Delta^2 = 4(a^2 b^2 + b^2 c^2 + c^2 a^2)$:
$$ \sin^2\omega = \frac{4\Delta^2}{a^2 b^2 + b^2 c^2 + c^2 a^2}, \qquad \csc^2\omega = \frac{1}{\sin^2 A} + \frac{1}{\sin^2 B} + \frac{1}{\sin^2 C}. $$
Bicentric Brocard Pair $(\Omega, \Omega')$: Homogeneous barycentrics $\Omega = \left(\frac{1}{b^2} : \frac{1}{c^2} : \frac{1}{a^2}\right)$ and $\Omega' = \left(\frac{1}{c^2} : \frac{1}{a^2} : \frac{1}{b^2}\right)$, whose midpoint $X(39)$ is the Brocard Midpoint on $[OK]$.
Seven-Point Brocard Circle: Circle on diameter $[OK]$ passing through $O, K, \Omega, \Omega'$ and Brocard’s First Triangle $A_1 = (a^2 : c^2 : b^2)$, $B_1 = (c^2 : b^2 : a^2)$, $C_1 = (b^2 : a^2 : c^2)$.
Antipodal Circumcircle Pair $(S_t, T_r)$: Parallels and perpendiculars from $A, B, C$ to $\triangle A_1 B_1 C_1$ concur by Thales’s theorem (Euclid
III.31) at diametrically opposite points on the circumcircle: Steiner’s Point $X(99) = \left(\frac{1}{b^2 - c^2} : \frac{1}{c^2 - a^2} : \frac{1}{a^2 - b^2}\right)$ and Tarry’s Point $X(98)$.
2. Verified Exact Ledger on $\triangle ABC$ $(15, 13, 14)$ (geometor.casey.brocard)¶
Constructed in construct_brocard_and_lemoine_circles() (0220.g1.svg):
Geometric Object |
Symbol / ETC Index |
Exact Rational Value |
|---|---|---|
Brocard Cotangent & Sine-Squared |
$\cot\omega, \; \sin^2\omega$ |
$\cot\omega = \frac{295}{168}, \quad \sin^2\omega = \frac{28224}{115249}$ |
First Brocard Point |
$\Omega$ |
$\left(\frac{697970}{115249}, \frac{397488}{115249}\right)$ |
Second Brocard Point |
$\Omega'$ |
$\left(\frac{684236}{115249}, \frac{529200}{115249}\right)$ |
Brocard’s First Triangle |
$A_1, \; B_1, \; C_1$ |
$\left(\frac{3589}{590}, \frac{1014}{295}\right), \; \left(\frac{3491}{590}, \frac{270}{59}\right), \; \left(7, \frac{1176}{295}\right)$ |
Brocard Circle (7 points) |
$(M_L, \frac{1}{2}|OK|)$ |
Centre $\left(\frac{1869}{295}, \frac{19143}{4720}\right)$, $R_{\text{Br}}^2 = \frac{397657}{891136}$ |
Steiner Point |
$S_t = X(99)$ |
$\left(\frac{17253}{2353}, -\frac{9396}{2353}\right) \in (O, A)$ |
Tarry Point |
$T_r = X(98)$ |
$\left(\frac{15689}{2353}, \frac{115233}{9412}\right) \in (O, A), \quad \frac{S_t + T_r}{2} = O$ |
de Longchamps Point |
$L = X(20) = 2O - H$ |
$\left(9, \frac{9}{2}\right)$ |