Midpoint

The center point (not to be confused with the center point ) is one of the special points of a triangle . It is obtained by connecting the centers of the three circles of the given triangle with the corresponding side centers. The connecting lines created in this way intersect at the midpoint. This property was proven in 1836 by the German mathematician Christian Heinrich von Nagel .

Triangle ABC with center point M (red), side centers (M _a , M _b , M _c ) and circle centers (J _A , J _B , J _C )

properties

• The center point is also the Lemoine point of the triangle, which is determined by the three circle centers.
• The center point lies on a straight line with the inscribed center point and the Lemoine point.
• The center point lies on a straight line with the center of gravity and the Gergonne point .
• The midpoint lies on a straight line with the vertical intersection and the Spieker point .
• The midpoint is the center of the Mandart inellipse .

Coordinates

Midpoint ( ) ${\displaystyle X_{9}}$
Trilinear coordinates	${\displaystyle (b+c-a)\,:\,(c+a-b)\,:\,(a+b-c)}$ ${\displaystyle =\,\cot {\frac {\alpha }{2}}\,:\,\cot {\frac {\beta }{2}}\,:\,\cot {\frac {\gamma }{2}}}$
Barycentric coordinates	${\displaystyle a(b+c-a)\,:\,b(c+a-b)\,:\,c(a+b-c)}$

Web links

• Eric W. Weisstein : midpoint . In: MathWorld (English).
• Center point - a visualization with GeoGebra