In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the incircle of the medial triangle

Medial triangle

The medial triangle of a triangle ABC is the triangle with vertices at the midpoints of the triangle's sides AB, AC and BC....

 of a triangle ABC is the Spieker circle. Its center, the Spieker center, is the center of mass of the boundary of triangle ABC as well as being the incenter of the medial triangle.

The Spieker center (X₁₀ according to Kimberling's enumeration

Encyclopedia of Triangle Centers

The Encyclopedia of Triangle Centers is an on-line list of more than 3,000 points or "centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville....

) is collinear with the incenter (X₁), centroid

Centroid

In geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...

 (X₂), and Nagel point

Nagel point

In geometry, the Nagel point is a point associated with any triangle. Given a triangle ABC, let TA, TB, and TC be the extouch points in which the A-excircle meets line BC, the B-excircle meets line CA, and C-excircle meets line AB, respectively...

(X₈). Indeed, the distances between pairs of the four points have constant ratios, regardless of the shape of triangle ABC, as indicated here:

External links

• Spieker Conic and generalization of Nagel line at Dynamic Geometry Sketches Generalizes Spieker circle and associated Nagel line.