Unit 6: Supplementary Chapter, Secs. II–VII & Misc. — Brocard Geometry, Tucker/Taylor Circles & Associated Points¶
John Casey — Supplementary Chapter on the Recent Elementary Geometry of the Triangle (5th ed., 1888 / 6th ed., 1892)
Section II — Figures Directly Similar, Brocard Points $\Omega, \Omega'$ & Brocard Angle $\omega$¶
Theory of Two Figures Directly Similar (Props. 1–2). Given any two directly similar plane figures, there exists a unique Double (Self-Corresponding) Point—the Centre of Spiral Similarity—about which a rotation through a fixed angle combined with a constant homothecy carries the first figure onto the second.
Section II, Propositions 3–5 (:person:`Brocard’s <brocard>` :term:`Points <brocard-points>` $\Omega, \Omega'$ & :term:`Brocard Angle <brocard-angle>` $\omega$). In any triangle △ABC:
The three circles described on $AB$ tangent to $BC$ at $B$, on $BC$ tangent to $CA$ at $C$, and on $CA$ tangent to $AB$ at $A$ concur at a unique interior point Ω (the First Brocard Point) such that
$$ \angle \Omega AB = \angle \Omega BC = \angle \Omega CA = \omega. $$
Cyclically reversing orientation, the three circles tangent at $A, B, C$ in the opposite sense concur at the Second Brocard Point Ω′ (isogonal conjugate of Ω) with the same common acute angle $\angle \Omega' BA = \angle \Omega' CB = \angle \Omega' AC = \omega$.
The Brocard Angle $\omega$ satisfies the fundamental cotangent and cosecant identities:
$$ \cot \omega = \cot A + \cot B + \cot C = \frac{a^2 + b^2 + c^2}{4\Delta}, \qquad \csc^2 \omega = \csc^2 A + \csc^2 B + \csc^2 C, $$
with sharp upper bound $\omega \le 30^\circ$ (equality iff $\triangle ABC$ is equilateral).
Section III (Part 2) & Section IV — Tucker’s Circles, Taylor’s Circle & Three Similar Figures¶
Section III, Propositions 2–5 (:person:`Tucker’s <tucker>` :term:`Circles & Taylor’s Circle <tucker-and-taylor-circles>`).
Tucker’s Six-Point Family: If a triangle $\triangle A'B'C'$ be homothetic to $\triangle ABC$ with homothetic centre at the Symmedian Point K, the six intersections of the sides of $\triangle A'B'C'$ (produced) with the non-corresponding sides of $\triangle ABC$ lie upon a single circle—a Tucker Circle—whose centre lies on line [O K] and whose antiparallel chords subtend constant angle with $OK$.
Special Tucker Circles: The Circumcircle (O A), Lemoine’s First Circle (Ml D1), and the Cosine Circle are all special members of the Tucker family.
:person:`Taylor’s <taylor>` Circle (1882): Let $D_a, D_b, D_c$ be the feet of the altitudes of $\triangle ABC$, and drop perpendiculars from each altitude foot upon the other two sides of the triangle. The six feet of these perpendiculars lie upon a single Tucker circle—Taylor’s Circle—whose centre is the incentre of the medial-orthic antiparallel triangle!
Section IV — General Theory of Three Similar Figures. Extending Section II to three directly similar figures constructed homologously upon the sides of $\triangle ABC$, their three pairwise centres of similarity lie on their Circle of Similitude, and homologous lines bound variable triangles constantly inversely similar, collapsing at a minimum-area configuration linked directly to Ω, Ω′, and K.
Section V — Brocard’s Circle, Brocard’s First Triangle & Second Proof of Feuerbach’s Theorem¶
Section V, Propositions 1–3 (:term:`The Seven-Point Brocard Circle <brocard-circle>` & Brocard’s First Triangle). Construct the circle (Ml O) having the segment joining the Circumcentre O and Symmedian Point K as diameter (centered at Ml $= X(182)$ of radius $\frac{1}{2}|OK| = \frac{1}{2} R \sqrt{1 - 3\tan^2\omega}$*). Let the pairs of Brocard rays* [B Om1] and [C Om2] meet at A1, [C Om1] and [A Om2] at B1, and [A Om1] and [B Om2] at C1, forming Brocard’s First Triangle △A1B1C1. Then:
All seven points O, K, Ω, Ω′, A1, B1, C1 lie concyclically upon the Brocard Circle (Ml O);
Ω and Ω′ are symmetric across the diameter [O K] with $\angle \Omega O K = \angle \Omega' O K = \omega$ and $\angle O \Omega K = \angle O \Omega' K = 90^\circ$;
Brocard’s First Triangle △A1B1C1 is inversely similar to △ABC with similarity ratio $\sin \omega$ and shares the same Centroid $G = X(2)$ as $\triangle ABC$!
Section V, Addendum (:person:`M’Clelland’s <mclelland>` Second Proof of Feuerbach’s Theorem, pp. 201–203). Without invoking circular inversion, Casey presents W. J. M'Clelland's elementary metric proof showing directly from the Euler-line projection of $O, I, N, H$ upon $BC$ that $|NI|^2 = (R_9 - r)^2$ and $|NI_a|^2 = (R_9 + r_a)^2$, providing a third independent verification of Feuerbach's Theorem!
Sections VI–VII & Miscellaneous Exercises — Harmonic Polygons, Steiner, Tarry, Neuberg & Longchamps¶
Section VI — Theory of :term:`Harmonic Polygons <harmonic-polygon>` (Casey, 1886; pp. 205–218). Generalizing the symmedian geometry of a triangle (every cyclic $3$-gon is harmonic!) to cyclic $n$-gons: a cyclic polygon $A_1 A_2 \dots A_n$ is harmonic when it is the circular-inversion or projective image of a regular $n$-gon—equivalently, when there exists an interior Lemoine point $K$ whose perpendicular distances to the $n$ sides are proportional to the side lengths, generating generalized Brocard points, Lemoine circles, and Cosine circles for any $n \ge 4$.
Section VII & Miscellaneous Exercises (:person:`Steiner’s <steiner>` Point $X(99)$, :person:`Tarry’s <tarry>` Point $X(98)$, :person:`Neuberg’s <neuberg>` Circles & :person:`de Longchamps’ <longchamps>` Point $X(20)$).
:term:`Steiner’s Point <steiner-and-tarry-points>` (:g:`St <0220.g1:St>` $= X(99)$): Straight lines drawn through the vertices A, B, C parallel respectively to the sides $B_1 C_1, C_1 A_1, A_1 B_1$ of Brocard’s First Triangle concur at a single point St lying on the Circumcircle (O A) (fourth intersection of the Circumcircle with Steiner’s Circumellipse).
:term:`Tarry’s Point <steiner-and-tarry-points>` (:g:`Tr <0220.g1:Tr>` $= X(98)$): Perpendiculars drawn through $A, B, C$ to the corresponding sides $B_1 C_1, C_1 A_1, A_1 B_1$ of Brocard’s First Triangle concur at Tr on the Circumcircle—diametrically opposite to St so that $\operatorname{midpoint}(S_t, T_r) = O$!
Neuberg’s Circles: The locus of the vertex $P$ of a variable triangle $\triangle PBC$ on fixed base $BC$ having the same Brocard angle $\omega$ as $\triangle ABC$ is a circle (an $A$-Neuberg Circle) orthogonal to the circle on $BC$.
de Longchamps’ Point & Circle (Misc. Ex. 146–158, p. 247): The reflection $L = 2 O - H = X(20)$ of the Orthocentre $H$ across the Circumcentre $O$ on the Euler Line is the Orthocentre of the anticomplementary triangle and centre of Longchamps’ radical circle.