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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.
The latter two can be done with a construction based on the intercept theorem. A slightly less elementary construction using these tools is based on the geometric mean theorem and will construct a segment of length from a constructed segment of length . It follows that every algebraically constructible number is geometrically constructible, by ...
In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ∠ ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid 's Elements . [ 1 ]
Intermediate value theorem ; Intercept theorem (Euclidean geometry) Intersecting chords theorem (Euclidean geometry) Intersecting secants theorem (Euclidean geometry) Intersection theorem (projective geometry) Inverse eigenvalues theorem (linear algebra) Inverse function theorem (vector calculus) Ionescu-Tulcea theorem (probability theory)
English: Diagram for a proof of the intercept theorem. Assume A C {\displaystyle AC} and B D {\displaystyle BD} are not parallel. Then the parallel line to A C {\displaystyle AC} through D {\displaystyle D} intersects S A {\displaystyle SA} in B 0 ≠ B {\displaystyle B_{0}\neq B} .
Intercept theorem; M. Midpoint theorem (conics) Mohr–Mascheroni theorem; N. Newton's theorem about ovals; P. Pasch's theorem; Poncelet–Steiner theorem ...
If for a homothety with center the image of a point is given (see diagram) then the image of a second point , which lies not on line can be constructed graphically using the intercept theorem: is the common point th two lines ¯ and ¯.
The extension of BO meets the half circle in R and due to Thales' theorem the line segment OR is the altitude of the right-angled triangle QNR. Hence the geometric mean theorem can be applied, which means that OR forms the side of a square OUSR with the same area as the rectangle BLNO and hence as the quarter circle. [7]