X591

1st VAN LAMOEN PERPENDICULAR BISECTORS POINT

Erect squares outwardly on the sides of triangle ABC. Two edges emanate from A; let P and Q be their endpoints. Let a’ be the perpendicular bisector of PQ, and define b’ and c’ cyclically. Then a’, b’, c’ concur in X591. See also X1991. (Floor van Lamoen, 1/4/2001, Hyacinthos #2123)
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 1991, 2 }: Centroid–Symmedian Line { 2, 6 }: Centroid–Symmedian Line { 372, 754 }: Central Line L(372, 754) { 637, 1152 }: Central Line L(637, 1152) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1991 = 2nd VAN LAMOEN PERPENDICULAR BISECTORS POINT (0.6545, 0.3258) X_2 = CENTROID (0.5374, 0.2000) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_372 = {X(3),X(6)}-HARMONIC CONJUGATE OF X(371) (0.7148, 0.4568) X_754 = EVEN (0, 4) INFINITY POINT (0.1795, -0.2651) X_637 = ANTICOMPLEMENT OF X(371) (0.4819, 0.1805) X_1152 = ISOGONAL CONJUGATE OF X(1132) (-0.9520, -2.2961) X_591 = 1st VAN LAMOEN PERPENDICULAR BISECTORS POINT (0.4203, 0.0741) A B C X1991 X2 X6 X372 X754 X637 X1152 X591 (a, b, c) = (5, 6, 7) | AB = 1

X591 (1st VAN LAMOEN PERPENDICULAR BISECTORS POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(bc[b^{2} + c^{2} - 2a^{2} + 4 \operatorname{area}(ABC)]\)
Search Value (6, 9, 13): \(7.573813236933\)
▸ Incident Central Lines4 lines

{2, 6}, {372, 754}, {488, 3071}, {637, 1152}

▸ Functional Relations3 relations

  • X591 = reflection of X1991 in X2

  • X591 = perspector of inner Vecten triangle and outer Vecten of outer Vecten triangle

  • X591 = centroid of {Ab, Ac, Bc, Ba, Ca, Cb} used in construction of 4th Lozada circle