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In a triangle, any arbitrary side can be considered the base. The two endpoints of the base are called base vertices and the corresponding angles are called base angles. The third vertex opposite the base is called the apex. The extended base of a triangle (a particular case of an extended side) is the line that contains the base.
An altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side. This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude. [23] The length of the altitude is the distance between the base and the vertex.
A proof from Euclid's Elements that, given a line segment, one may construct an equilateral triangle that includes the segment as one of its sides: an equilateral triangle ΑΒΓ is made by drawing circles Δ and Ε centered on the points Α and Β, and taking one intersection of the circles as the third vertex of the triangle.
These attitudes are specified with two angles. For a line, these angles are called the trend and the plunge. The trend is the compass direction of the line, and the plunge is the downward angle it makes with a horizontal plane. [15] For a plane, the two angles are called its strike (angle) and its dip (angle).
If two points A, B of a line a lie in a plane α, then every point of a lies in α. In this case we say: “The line a lies in the plane α,” etc. If two planes α, β have a point A in common, then they have at least a second point B in common. There exist at least four points not lying in a plane. II. Order
For any triangle, and, in particular, any right triangle, there is exactly one circle containing all three vertices of the triangle. (Sketch of proof. The locus of points equidistant from two given points is a straight line that is called the perpendicular bisector of the line segment connecting the points.
The circle meets the angle at two points: one on each leg. Using each of these points as a center, draw two circles of the same size. The intersection of the circles (two points) determines a line that is the angle bisector. The proof of the correctness of this construction is fairly intuitive, relying on the symmetry of the problem.
Each of a triangle's excircles (orange) is tangent to one of the triangle's sides and to the other two extended sides. In plane geometry, an extended side or sideline of a polygon is the line that contains one side of the polygon. The extension of a finite side (a line segment) into an infinite line arises in various contexts.