X1991
2nd VAN LAMOEN PERPENDICULAR BISECTORS POINT
Erect squares inwardly 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
X1991 . See
X591 for the 1st Van Lamoen perpendicular bisectors point, constructed from outwardly drawn squares.
5, 6, 7
3, 4, 5
Kepler
Kimberling
< 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) { 591, 2 }: Centroid–Symmedian Line { 2, 6 }: Centroid–Symmedian Line { 371, 754 }: Central Line L(371, 754) { 638, 1151 }: Central Line L(638, 1151)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_591 = 1st VAN LAMOEN PERPENDICULAR BISECTORS POINT (0.4203, 0.0741) X_2 = CENTROID (0.5374, 0.2000) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_371 = KENMOTU POINT (CONGRUENT SQUARES POINT) (0.5652, 0.2097) X_754 = EVEN (0, 4) INFINITY POINT (0.1795, -0.2651) X_638 = ANTICOMPLEMENT OF X(372) (0.1827, -0.3137) X_1151 = ISOGONAL CONJUGATE OF X(1131) (0.5483, 0.1819) X_1991 = 2nd VAN LAMOEN PERPENDICULAR BISECTORS POINT (0.6545, 0.3258)
A
B
C
X591
X2
X6
X371
X754
X638
X1151
X1991
(a, b, c) = (5, 6, 7) | AB = 1
< 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) { 591, 2 }: Centroid–Symmedian Line { 2, 6 }: Centroid–Symmedian Line { 371, 754 }: Central Line L(371, 754) { 638, 1151 }: Central Line L(638, 1151)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_591 = 1st VAN LAMOEN PERPENDICULAR BISECTORS POINT (0.3522, -0.0067) X_2 = CENTROID (0.5467, 0.1600) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6400, 0.2400) X_371 = KENMOTU POINT (CONGRUENT SQUARES POINT) (0.5946, 0.1622) X_754 = EVEN (0, 4) INFINITY POINT (0.1642, -0.2694) X_638 = ANTICOMPLEMENT OF X(372) (0.1015, -0.4431) X_1151 = ISOGONAL CONJUGATE OF X(1131) (0.5714, 0.1224) X_1991 = 2nd VAN LAMOEN PERPENDICULAR BISECTORS POINT (0.7411, 0.3267)
A
B
C
X591
X2
X6
X371
X754
X638
X1151
X1991
(a, b, c) = (3, 4, 5) | AB = 1
< 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) { 591, 2 }: Centroid–Symmedian Line { 2, 6 }: Centroid–Symmedian Line { 371, 754 }: Central Line L(371, 754) { 638, 1151 }: Central Line L(638, 1151)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_591 = 1st VAN LAMOEN PERPENDICULAR BISECTORS POINT (0.3774, -0.0047) X_2 = CENTROID (0.5393, 0.1620) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6180, 0.2429) X_371 = KENMOTU POINT (CONGRUENT SQUARES POINT) (0.5794, 0.1635) X_754 = EVEN (0, 4) INFINITY POINT (0.1979, -0.3007) X_638 = ANTICOMPLEMENT OF X(372) (0.1589, -0.4592) X_1151 = ISOGONAL CONJUGATE OF X(1131) (0.5599, 0.1232) X_1991 = 2nd VAN LAMOEN PERPENDICULAR BISECTORS POINT (0.7013, 0.3286)
A
B
C
X591
X2
X6
X371
X754
X638
X1151
X1991
Kepler (1, √φ, φ) | AB = 1
< 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) { 591, 2 }: Centroid–Symmedian Line { 2, 6 }: Centroid–Symmedian Line { 371, 754 }: Central Line L(371, 754) { 638, 1151 }: Central Line L(638, 1151)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_591 = 1st VAN LAMOEN PERPENDICULAR BISECTORS POINT (0.2031, -0.1246) X_2 = CENTROID (0.5444, 0.0934) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6573, 0.1655) X_371 = KENMOTU POINT (CONGRUENT SQUARES POINT) (0.6182, 0.0560) X_754 = EVEN (0, 4) INFINITY POINT (0.0698, -0.3596) X_638 = ANTICOMPLEMENT OF X(372) (0.1628, -0.4864) X_1151 = ISOGONAL CONJUGATE OF X(1131) (0.5947, -0.0098) X_1991 = 2nd VAN LAMOEN PERPENDICULAR BISECTORS POINT (0.8857, 0.3113)
A
B
C
X591
X2
X6
X371
X754
X638
X1151
X1991
Kimberling (6, 9, 13) | AB = 1
X1991 (2nd VAN LAMOEN PERPENDICULAR BISECTORS POINT ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(bc[b^{2} + c^{2} - 2a^{2} - 4 \operatorname{area}(ABC)]\)
Search Value (6, 9, 13): \(-2.315075651956\)
▸ Incident Central Lines4 lines
{2 6 371 754 487 3070 638 1151
▸ Functional Relations3 relations
X1991 X591 X2
X1991
X1991
