Viviani's theorem
In an equilateral triangle, the perpendicular distances from any interior point to its three sides sum to the triangle’s altitude.
Triangle area: The area of a triangle is half the product of a base and its corresponding height. Joining the interior point to the vertices partitions the triangle into three smaller triangles whose areas add.
Perpendicular distance: The shortest distance from a point to a line, measured along a perpendicular. Each term in Viviani’s sum is the perpendicular distance to one side.
Centroid: The point where a triangle’s three medians intersect; it divides each median in a two-to-one ratio. In an equilateral triangle, the centroid is equally distant from all three sides, so each distance is one-third of the altitude.
Vincenzo Viviani: An Italian mathematician and scientist who lived from 1622 to 1703 and worked with Galileo. Viviani’s name is attached to the equilateral-triangle distance theorem.
Altitude: An altitude is a perpendicular segment from a vertex to the line containing the opposite side. The theorem’s constant sum is exactly the equilateral triangle’s altitude.
Interior point: A point inside a geometric figure, excluding its boundary. The theorem applies to points inside the triangle, where all three side distances are positive.
Incenter: The point where a triangle’s angle bisectors intersect, equidistant from its sides. For an equilateral triangle, its incenter is the centroid and realizes the equal-distance case.
Galileo Galilei: An Italian astronomer, physicist, and mathematician who lived from 1564 to 1642. Viviani studied and worked with Galileo, linking the theorem’s namesake to a major scientific circle.
Equilateral triangle: A triangle with three equal sides and three equal angles of 60 degrees. Equal side lengths let the three area terms share one common base.
Base and height: A base and its corresponding height are a chosen side of a figure and the perpendicular distance to it. Each smaller triangle uses one side as a base and the interior point’s distance as its height.