mediansidepunkt

The mediansidepunkt is a geometric concept describing the point at which a triangle’s median meets its opposite side. In standard terms this is simply the midpoint of that side. For a triangle with vertices A, B, and C, the mediansidepunkt on side BC is the point M on BC such that BM = MC. There are three mediansidepunkts, one on each side, corresponding to the three medians.

The term can be used to emphasize the endpoint of a median on a side, as distinct from the centroid, which is the intersection point of all three medians inside the triangle. The three mediansidepunkts are the vertices of the medial triangle, a triangle formed by connecting the midpoints of the original triangle’s sides. The medial triangle is similar to the original triangle with half its side lengths.

Properties include that the mediansidepunkts define a triangle whose sides are parallel to the original triangle’s sides. The segment joining two mediansidepunkts is parallel to the third side of the original triangle. The centroid, located at the intersection of the medians, divides each median in a 2:1 ratio, counting from the vertex toward the side midpoint.

Construction typically involves finding the midpoint of a given side by standard compass-and-straightedge methods. In practice, mediansidepunkts are often used in discussions of midpoint geometry, the medial triangle, and coordinate or barycentric approaches to triangle problems.

Etymology: mediansidepunkt appears to be a coinage combining median and a term for a point on a side; in mainstream geometry the standard terms are “side midpoint” or “midpoint of a side.” See also centroid, medial triangle, side midpoint.