Triangle geometry

Triangle geometry studies the properties of triangles and the points, lines, circles, and relationships associated with them.

Triangle: A polygon with three sides, three vertices, and three interior angles. Its side lengths and angles define the basic objects of triangle geometry.
Circumcenter: The point equidistant from a triangle's three vertices and center of its circumcircle. It is obtained by intersecting the perpendicular bisectors of the sides.
Morley's trisector theorem: A theorem stating that adjacent angle trisectors of any triangle form an equilateral triangle. It produces a hidden regular figure from the angles of an arbitrary triangle.
Euclidean geometry: The geometry of points, lines, distances, and angles governed by Euclid's parallel postulate. Most classical triangle centers and circle theorems assume this geometry.
Trigonometry: The study of relationships between angles and side lengths, especially in triangles. Its laws calculate unknown sides and angles from partial triangle data.
Ceva's theorem: A criterion stating when three lines through a triangle's vertices are concurrent. It tests whether cevians meet at one point using ratios on the opposite sides.
Incenter: The point equidistant from a triangle's three sidelines and center of its incircle. The angle bisectors meet there, defining the circle tangent to all three sides.
Napoleon's theorem: A theorem stating that equilateral triangles erected externally on a triangle's sides have collinear centroids. It shows how an added construction creates a rigid alignment of centers.
Hyperbolic geometry: A non-Euclidean geometry in which parallel lines through a point are not unique and triangle angle sums are below 180 degrees. Its triangles do not obey the Euclidean angle-sum and center results unchanged.
Computational geometry: The study of algorithms for solving geometric problems on finite collections of objects. Triangle predicates and triangulations are basic computational geometry tools.

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