## X(140) (MIDPOINT OF X(3) AND X(5))

 Interactive Applet

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This applet was built with the free and multiplatform dynamic geometry software C.a.R..

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           2 cos A + cos(B - C) : 2 cos B + cos(C - A) : 2 cos C + cos(A - B)
= cos A + 2 sin B sin C : cos B + 2 sin C sin A : cos C + 2 sin A sin B
= 3 cos A + 2 cos B cos C : 3 cos B + 2 cos C cos A : 3 cos C + 2 cos A cos B
= f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = bc[b cos(C - A) + c cos(B - A)]

Barycentrics    g(a,b,c) : g(b,c,a) : g(c,a,b) where g(a,b,c) = b cos(C - A) + c cos(B - A)

X(140) lies on the Euler line
X(140) = crosspoint of the two Napoleon points
X(140) = X(5)-of-medial triangle

X(140) lies on these lines:
2,3    10,214    11,35    12,36    15,18    16,17    39,230    54,252    55,496    56,495    61,395    62,396    95,340    125,128    141,182    143,511    195,323    298,628    299,627    302,633    303,634    343,569    371,615    372,590    524,575    576,597    601,748    602,750    618,630    619,629

X(140) is the {X(2),X(3)}-harmonic conjugate of X(5).

X(140) = midpoint of X(I) and X(J) for these (I,J): (3,5), (141,182)
X(140) = reflection of X(I) in X(J) for these (I,J): (546,5), (547,2), (548,3)
X(140) = inverse-in-orthocentroidal-circle of X(1656)
X(140) = isogonal conjugate of X(1173)
X(140) = complement of X(5)
X(140) = complementary conjugate of X(1209)
X(140) = X(2)-Ceva conjugate of X(233)
X(140) = crosspoint of X(I) and X(J) for these (I,J): (2,95), (17,18)
X(140) = crosssum of X(I) and X(J) for these (I,J): (6,51), (61,62)

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.