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Equivalently, an inscribed angle is defined by two chords of the circle sharing an endpoint. The inscribed angle theorem relates the measure of an inscribed angle to that of the central angle subtending the same arc. The inscribed angle theorem appears as Proposition 20 in Book 3 of Euclid's Elements.
An alternative way uses the inscribed angle theorem for parabolas. In the following, the angle of two lines will be measured by the difference of the slopes of the line with respect to the directrix of the parabola.
A parabolic segment is the region bounded by a parabola and line. To find the area of a parabolic segment, Archimedes considers a certain inscribed triangle. The base of this triangle is the given chord of the parabola, and the third vertex is the point on the parabola such that the tangent to the parabola at that point is parallel to the chord.
Inscribed angle theorem for an ellipse. Like a circle, such an ellipse is determined by three points not on a line. For this family of ellipses, one introduces the following q-analog angle measure, which is not a function of the usual angle measure θ: [16] [17]
Pascal's theorem is the polar reciprocal and projective dual of Brianchon's theorem. It was formulated by Blaise Pascal in a note written in 1639 when he was 16 years old and published the following year as a broadside titled "Essay pour les coniques. Par B. P." [1] Pascal's theorem is a special case of the Cayley–Bacharach theorem.
From the inscribed angle theorem one gets: The intersection points of corresponding lines form a circle. Examples of commonly used fields are the real numbers R {\displaystyle \mathbb {R} } , the rational numbers Q {\displaystyle \mathbb {Q} } or the complex numbers C {\displaystyle \mathbb {C} } .
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The similarity yields an equation for ratios which is equivalent to the equation of the theorem given above: = | | | | = | | | | Next to the intersecting chords theorem and the tangent-secant theorem , the intersecting secants theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle ...