X3310–X3414

Incircle Transforms T(U), Trigonometric Offset Points & Kiepert Cusps

Barycentric products of perpendicular directions (X3310), incircle transforms T(U) lying on the incircle (X3318–X3328), Darboux-cubic reflections and conjugates (X3341–X3358), trigonometric phase-shifted centers cos/sin/sec/csc/tan/cot(A ± θ) (X3364–X3409), and the Brocard-Kiepert-Fermat cusps (X3411–X3414).
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) ( A R_b ): Euclid I.22 Guide Circle (A) ( B R_a ): Euclid I.22 Guide Circle (B) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) circumcircle: Circumcircle of △ABC incircle: Incircle of △ABC A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) R_b = Radius Point R_b (0.8571, 0.0000) R_a = Radius Point R_a (0.2857, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1 = INCENTER (0.5714, 0.2333) X_3310 = BARYCENTRIC PRODUCT X(513)*X(517) (1.1250, 0.6124) X_3311 = INTERSECTION OF LINES X(3)X(6) AND X(4)X(1131) (0.5789, 0.2324) X_3312 = INTERSECTION OF LINES X(3)X(6) AND X(4)X(1132) (0.6365, 0.3274) X_3313 = INTERSECTION OF LINES X(3)X(6) AND X(66)X(69) (0.3845, -0.0887) X_3314 = INTERSECTION OF LINES X(2)X(6) AND X(76)X(115) (0.4770, 0.1350) X_3315 = INTERSECTION OF LINES X(1)X(88) AND X(105)X(110) (0.6571, 0.1866) X_3318 = INCIRCLE TRANSFORM T(X(603)) (0.5568, 0.0005) X_3319 = INCIRCLE TRANSFORM T(X(654)) (0.7941, 0.3030) X_3320 = INCIRCLE TRANSFORM T(X(656)) (0.5788, 0.4665) X_3321 = INCIRCLE TRANSFORM T(X(663)) (0.4744, 0.4455) X_3323 = INCIRCLE TRANSFORM T(X(692)) (0.3587, 0.3291) X_3325 = INCIRCLE TRANSFORM T(X(896)) (0.7947, 0.1657) X_3326 = INCIRCLE TRANSFORM T(X(909)) (0.3488, 0.1636) X_3360 = SS(a → bc) OF X(20) (0.9152, 0.4425) X_3367 = CSC(A - 5π/12) POINT (0.5942, 0.8035) X_3372 = SIN(A - 3π/8) POINT (0.1555, -0.4670) A B C X3 X1 X3310 X3311 X3312 X3313 X3314 X3315 X3318 X3319 X3320 X3321 X3323 X3325 X3326 X3360 X3367 X3372 (a, b, c) = (5, 6, 7) | AB = 1

X3310–X3414 (Incircle Transforms T(U), Trigonometric Offset Points & Kiepert Cusps) — 16 representative centers (from 105 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Barycentric Products of Perpendicular Directions

Suppose P and U are points on the line at infinity, given in barycentric coordinates by P =\(p : q : r\) and U =\(u : v : w\). We call P and U perpendicular directions if for every point X not in the line at infinity, the lines XP and XU are perpendicular. (It is well known that if P and U are an antipodal pair on the circumcircle, then their isogonal conjugates are perpendicular directions.)

Theorem: If P and U are perpendicular directions, then their barycentric product lies on the orthic axis.

An outline of a proof follows. For given P on the line at infinity, the direction perpendicular to P is given by

a/\(d_{1}\) : b/\(d_{2}\) : c/\(d_{3}\),
where
\(d_{1}\) = \(a^{2}\)qr cos B cos C - \(p(b_{2}\)r + \(c_{2}\)q) cos A
\(d_{2}\) = \(b^{2}\)rp cos C cos A - \(q(c_{2}\)p + \(a_{2}\)r) cos B
\(d_{3}\) = \(c^{2}\)pq cos A cos B - \(r(a_{2}\)q + \(b_{2}\)p) cos C,

and the barycentric product P*U is the point \(x : y : z\) given by

ap/\(d_{1}\) : bq/\(d_{2}\) : cr/\(d_{3}\).

For P to lie on the line at infinity means that r = - p - q. Substitution for r, a computer quickly shows that

x cot A + y cot B + z cot C = 0,

as desired.

The orthic axis is perpendicular to the Euler line. For this and other properties, visit MathWorld. The appearance of (I, J, K) in the following list means that X(K) is the barycentric product of perpendicular directions Xi and Xj.

(30,1511,1637)
(511,523,2491)
(514,516,676)
(2574,2575,647)
(513,517,3310)

Group Preamble — Points on the Incircle (Incircle transforms)

Suppose that U =\(u : v : w\) (trilinears) is a point other than the symmedian point, X6. Then the incircle transform of :g:`U <U*>` is the point.

T(U) = a (a - b + c) (a + b - c) (c v - b w)^2 : : ,

which lies on the incircle. Geometrically, T(U) is the Brisse transform of the psi transform of X6 and U. (Brisse transform is described just before X1254, and psi transform, also called the “Psi mapping” elsewhere in ETC, and TCCT, p. 80, is defined in the preamble just before X98.

X_3333 — POHOATA POINT

X3333POHOATA POINT

Let R be the circumradius, r the inradius, and \(r_{A}\) the radius of the A-excircle. The trilinears given above are equivalent to 2R - \(r_{A}\) : 2R - \(r_{B}\) : 2R - \(r_{C}\). The 1st trilinear representation
X_3334 — PERSPECTOR OF TRIANGLES T1T2T3 AND J1J2J3

X3334PERSPECTOR OF TRIANGLES T1T2T3 AND J1J2J3

The equilateral triangles T1T2T3 and J1J2J3 are defined just before X3272. The points X3334 and X3335, contributed April 16, 2008 by Peter Moses, complete a list (just before X3272 of perspectors associated with equilateral triangles.
X_3361 — INTERSECTION OF LINES X_1X_3 AND X_7X_1125

X3361INTERSECTION OF LINES X1X_3 AND X7X_1125

Let OA be the circle tangent to side BC at its midpoint and to the circumcircle on the side ob BC opposite A. Define OB and OC cyclically. Let A' be the insimilicanter of OB and OC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X3361. See the reference at X1001.
X_3363 — X_6 OF PEDAL TRIANGLE OF X_2

X3363OF PEDAL TRIANGLE OF X2

X3363 is the symmedian point of the pedal triangle of the centroid. A second construction uses the midpoint X597 of the symmedian point and centroid and the midpoint X115 of the two Fermat points, X13 and X14; specifically, X3363 is the point in which the line X115X597 meets the Euler line. Contributed by Po-chieh Chen and Shao-cheng Liu, May 29, 2008.
X_3364 — COS(A - 5π/12) POINT

X3364COS(A - 5π/12) POINT

X3364 and related points occupy Bernard Gibert’s Table 38 concerning these three loci: Brocard axis, Kiepert hyperbola, and the cubic K457. Trilinear equations for these curves, of degrees 1,2,3, respectively, in the variables a,b,c, are given as follows:

Curve Trilinears Brocard axis \(\cos(A + t) : \cos(B + t) : \cos(C + t)\) Kiepert hyperbola \(\sec(A + t) : \sec(B + t) : \sec(C + t)\) K457 \(\tan(A + t) : \tan(B + t) : \tan(C + t)\)

X_3411 — 1st BROCARD-KIEPERT-FERMAT CUSP

X34111st BROCARD-KIEPERT-FERMAT CUSP

X3411 and X3412 occur in Bernard Gibert’s study of the Brocard-Kiepert quartic, which has three cusps: X20 with Euler line as tangent; X3411 with line X5X13 as tangent, and X3412 with line X5X14 as tangent.