Short answers, pulled from the story.
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. Its corners are called vertices and its sides are called edges, and its three internal angles always sum to a straight angle of 180 degrees or pi radians.
By side length, a triangle is equilateral when all three sides are equal, isosceles when two sides are equal, and scalene when all three differ. By angle, it is a right triangle when one angle is a right angle, acute when all angles are smaller, and obtuse when one angle is larger. These definitions date back at least to Euclid.
Triangles are rigid because specifying the lengths of all three sides determines the angles, so the shape will not collapse unless its sides bend, stretch, or break. Structural quadrilaterals are often braced with a diagonal that splits them into two rigid triangles, and engineering uses tetrahedral trusses for cantilevering.
A triangle has a circumcenter at the center of the circumcircle, an incenter at the center of the incircle, an orthocenter where its altitudes meet, and a centroid where its medians meet. The orthocenter, the nine-point circle's center, the centroid, and the circumcenter all lie on Euler's line.
In Euclidean space the interior angles of a triangle always add up to 180 degrees, a fact equivalent to Euclid's parallel postulate. On a negatively curved surface a hyperbolic triangle sums to less than 180 degrees, and on a sphere a spherical triangle sums to more than 180 degrees.
One of the oldest methods takes half the product of a base and its corresponding altitude. Heron's formula, named after Heron of Alexandria, finds the area from the three side lengths using the semiperimeter, and the shoelace formula computes it from the Cartesian coordinates of the vertices.