## X(182) (MIDPOINT OF BROCARD DIAMETER)

 Interactive Applet

You can move the points A, B and C (click on the point and drag it).
Press the keys “+” and “−” to zoom in or zoom out the visualization window and use the arrow keys to translate it.

You can also construct all centers related with this one (as described in ETC) using the “Run Macro Tool”. To do this, click on the icon , select the center name from the list and, then, click on the vertices A, B and C successively.

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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           cos(A - ω) : cos(B - ω) : cos(C - ω)
=cos A + sin A tan ω : cos B + sin B tan ω : cos C + sin C tan ω
= sin A - sin(A - 2ω) : sin B - sin(B - 2ω) : sin C - sin(C - 2ω)
= cos A + cos(A - 2ω) : cos B + cos(B - 2ω) : cos C + cos(C - 2ω) (cf., X(39))

Barycentrics    sin A cos(A - ω) : sin B cos(B - ω) : sin C cos(C - ω)

X(182) is the midpoint of the Brocard diameter (the segment X(3)-to-X(6)); also the center of the 1st Lemoine circle, and the center of the Brocard circle.

X(182) lies on these lines:
1,983    2,98    3,6    4,83    5,206    10,1678    22,51    24,1843    30,597    36,1469    40,1700    54,69    55,613    56,611    111,353    140,141    171,1397    373,1495    474,1437    517,1386    518,1385    524,549    692,1001    727,1293    729,1296    952,996    1676,1677

X(182) is the {X(371),X(372)}-harmonic conjugate of X(39).

X(182) = midpoint of X(3) and X(6)
X(182) = reflection of X(I) in X(J) for these (I,J): (6,575), (141,140), (576,6)
X(182) = isogonal conjugate of X(262)
X(182) = isotomic conjugate of X(327)
X(182) = complement of X(1352)

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