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Equivalently, a line segment is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points. In geometry, one might define point B to be between two other points A and C, if the distance | AB | added to the distance | BC | is equal to the distance | AC |.
Let AB and BC be two segments of a line a which have no points in common aside from the point B, and, furthermore, let A′B′ and B′C′ be two segments of the same or of another line a′ having, likewise, no point other than B′ in common. Then, if AB ≅ A′B′ and BC ≅ B′C′, we have AC ≅ A′C′.
In Euclidean geometry, the intersecting chords theorem, or just the chord theorem, is a statement that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths of the line segments on each chord are equal.
In an axiomatic formulation of Euclidean geometry, such as that of Hilbert (modern mathematicians added to Euclid's original axioms to fill perceived logical gaps), [1]: 108 a line is stated to have certain properties that relate it to other lines and points. For example, for any two distinct points, there is a unique line containing them, and ...
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.
Menelaus's theorem, case 1: line DEF passes inside triangle ABC. In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle ABC, and a transversal line that crosses BC, AC, AB at points D, E, F respectively, with D, E, F distinct from A, B, C. A ...
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