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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.
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.
The Crossbar Theorem says ray AD intersects segment BC. In geometry, the crossbar theorem states that if ray AD is between ray AC and ray AB, then ray AD intersects line segment BC. [1] This result is one of the deeper results in axiomatic plane geometry. [2]
Diagram for geometric proof. This proof is valid only if the line is not horizontal or vertical. [5] Drop a perpendicular from the point P with coordinates (x 0, y 0) to the line with equation Ax + By + C = 0. Label the foot of the perpendicular R. Draw the vertical line through P and label its intersection with the given line S.
Here, p is the (positive) length of the line segment perpendicular to the line and delimited by the origin and the line, and is the (oriented) angle from the x-axis to this segment. It may be useful to express the equation in terms of the angle α = φ + π / 2 {\displaystyle \alpha =\varphi +\pi /2} between the x -axis and the line.
In Euclidean geometry, the intersecting secants theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle.
The midpoint theorem, midsegment theorem, or midline theorem states that if the midpoints of two sides of a triangle are connected, then the resulting line segment will be parallel to the third side and have half of its length.
Proof: Assume the derivative of at every interior point of the interval exists and is zero. Let ( a , b ) {\displaystyle (a,b)} be an arbitrary open interval in I {\displaystyle I} . By the mean value theorem, there exists a point c {\displaystyle c} in ( a , b ) {\displaystyle (a,b)} such that