
Area and barycenter
Barycentric coordinates offer a fresh approach to Routh's theorem and its applications to several results in Euclidean geometry.


Barycentric coordinates offer a fresh approach to Routh's theorem and its applications to several results in Euclidean geometry.


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Points and lines in the plane are dual notions: theorems about collinear points correspond to theorems about concurrent lines. This duality can be defined geometrically. It even extends to space, through coplanarity.

Beyond the centroid, the orthocenter and the centers of the two circles familiar from school geometry, thousands of points can be associated with the three vertices of a triangle. These myriad points lie on hundreds of cubics with remarkable properties.

The midpoint of two points and the center of gravity of a triangle are familiar concepts from middle school onward. What do they have in common? How can they be generalized? In the early 19th century, a new approach to geometry put them on a firm mathematical footing through the concept of the barycenter.

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