Gergonne point

The Gergonne point of a triangle (named after the French mathematician Joseph Diaz Gergonne ) is an excellent point inside a triangle . The inscribed circle of the triangle has the center and touches the sides of the triangle at points , and . Gergonne showed that the connecting lines between these points of contact and the opposite corner of the triangle intersect at one point, the Gergonne point . ${\displaystyle ABC}$ ${\displaystyle M}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Z}$ ${\displaystyle G}$

The fact that these lines intersect at a point follows from etc. and Ceva's theorem . ${\displaystyle {\overline {AZ}}={\overline {AY}}}$

properties

• The Gergonne point lies on a straight line with the center of gravity and the midpoint (in that order).
• Gergonne point and Nagel point are isotomically conjugated .

Coordinates

Gergonne point ( ) ${\displaystyle X_{7}}$
Trilinear coordinates	${\displaystyle {\frac {bc}{b+c-a}}\,:\,{\frac {ca}{c+a-b}}\,:\,{\frac {ab}{a+b-c}}}$ ${\displaystyle =\sec ^{2}{\frac {\alpha }{2}}\,:\,\sec ^{2}{\frac {\beta }{2}}\,:\,\sec ^{2}{\frac {\gamma }{2}}}$
Barycentric coordinates	${\displaystyle {\frac {1}{b+c-a}}\,:\,{\frac {1}{c+a-b}}\,:\,{\frac {1}{a+b-c}}}$

literature

• Peter Baptist: Historical Notes on Gergonne and Nagel Points. In: Sudhoffs Archiv , 71, 1987, 2, pp. 230-233

Web links

• Full proof. (English)
• Eric W. Weisstein : Gergonne Point . In: MathWorld (English).
• Gergonne point - visualization with GeoGebra