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With respect to the AB ray, the AD ray is called the opposite ray. Thus, we would say that two different points, A and B, define a line and a decomposition of this line into the disjoint union of an open segment (A, B) and two rays, BC and AD (the point D is not drawn in the diagram, but is to the left of A on the line AB). These are not ...
A triangle's altitudes run from each vertex and meet the opposite side at a right angle. The point where the three altitudes meet is the orthocenter. Angle bisectors are rays running from each vertex of the triangle and bisecting the associated angle. They all meet at the incenter.
In geometry, two lines and are antiparallel with respect to a given line if they each make congruent angles with in opposite senses.More generally, lines and are antiparallel with respect to another pair of lines and if they are antiparallel with respect to the angle bisector of and .
Let E be a point on the line s on the opposite side of A from C. Take A' on CB' so that A'B' = AB. Through A' draw a line s' (A'E') on the side closer to E, so that the angle B'A'E' is the same as angle BAE. Then s' meets s in an ordinary point D'. Construct a point D on ray AE so that AD = A'D'. Then D' ≠ D.
The concept of antipodal points is generalized to spheres of any dimension: two points on the sphere are antipodal if they are opposite through the centre.Each line through the centre intersects the sphere in two points, one for each ray emanating from the centre, and these two points are antipodal.
This result is one of the deeper results in axiomatic plane geometry. [2] It is often used in proofs to justify the statement that a line through a vertex of a triangle lying inside the triangle meets the side of the triangle opposite that vertex. This property was often used by Euclid in his proofs without explicit justification.
If the internal bisector of angle A in triangle ABC has length and if this bisector divides the side opposite A into segments of lengths m and n, then [3]: p.70 + = where b and c are the side lengths opposite vertices B and C; and the side opposite A is divided in the proportion b:c.
The intercept theorem, also known as Thales's theorem, basic proportionality theorem or side splitter theorem, is an important theorem in elementary geometry about the ratios of various line segments that are created if two rays with a common starting point are intercepted by a pair of parallels.