X370
EQUILATERAL CEVIAN TRIANGLE POINT
A point
P is an equilateral
cevian triangle point if the cevian triangle of
P is equilateral. Jiang Huanxin introduced this notion in 1997.
5, 6, 7
3, 4, 5
Kepler
Kimberling
< A B C >: Reference Triangle △ABC < P_a P_b P_c >: Cevian Triangle △P_a P_b P_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) [ A P_a ]: A-Cevian A–P_a [ B P_b ]: B-Cevian B–P_b [ C P_c ]: C-Cevian C–P_c
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_370 = EQUILATERAL CEVIAN TRIANGLE POINT (0.5771, 0.2488)
A
B
C
Pa
Pb
Pc
X370
(a, b, c) = (5, 6, 7) | AB = 1
< A B C >: Reference Triangle △ABC < P_a P_b P_c >: Cevian Triangle △P_a P_b P_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) [ A P_a ]: A-Cevian A–P_a [ B P_b ]: B-Cevian B–P_b [ C P_c ]: C-Cevian C–P_c
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_370 = EQUILATERAL CEVIAN TRIANGLE POINT (0.6064, 0.2232)
A
B
C
Pa
Pb
Pc
X370
(a, b, c) = (3, 4, 5) | AB = 1
< A B C >: Reference Triangle △ABC < P_a P_b P_c >: Cevian Triangle △P_a P_b P_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) [ A P_a ]: A-Cevian A–P_a [ B P_b ]: B-Cevian B–P_b [ C P_c ]: C-Cevian C–P_c
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_370 = EQUILATERAL CEVIAN TRIANGLE POINT (0.5895, 0.2258)
A
B
C
Pa
Pb
Pc
X370
Kepler (1, √φ, φ) | AB = 1
< A B C >: Reference Triangle △ABC < P_a P_b P_c >: Cevian Triangle △P_a P_b P_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) [ A P_a ]: A-Cevian A–P_a [ B P_b ]: B-Cevian B–P_b [ C P_c ]: C-Cevian C–P_c
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_370 = EQUILATERAL CEVIAN TRIANGLE POINT (0.6198, 0.1666)
A
B
C
Pa
Pb
Pc
X370
Kimberling (6, 9, 13) | AB = 1
X370 (EQUILATERAL CEVIAN TRIANGLE POINT ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Constructed Elements
Derived Triangle: Cevian Triangle △Pa Pb Pc — vertices Pa , Pb , Pc (Traces of cevians through target center on BC, CA, AB)
Concurrence Lines: A–Pa , B–Pb , C–Pc concurring at X370
Trilinears: \((see below)\)
Barycentrics: \((see below)\)
Search Value (6, 9, 13): \(1.277501263430\)
X370
Notes
Jean-Pierre Ehrmann notes (11/6/02) that the normalized barycentric coordinates (x,y,z) of X370
y(1 - y)\(S_{B}\) + z(1 - z)\(S_{C}\) = x(1 + x)F
z(1 - z)\(S_{C}\) + x(1 - x)\(S_{A}\) = y(1 + y)F
x(1 - x)\(S_{A}\) + y(1 - y)\(S_{B}\) = z(1 + z)F
x + y + z = 1,
where \(S_{A}\) = (\(b^{2}\) + \(c^{2}\) - \(a^{2})\) /2; \(S_{B}\) , \(S_{C}\) are defined cyclically, F = [2 area(ABC )]/\(\sqrt{3}\) , and x>0, y>0, z>0.
Jiang Huanxin and David Goering, Problem 10358* and Solution, “Equilateral cevian triangles,” American Mathematical Monthly 104 (1997) 567-570 [proposed 1994].
