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As a consequence of the theorem, opposite angles of cyclic quadrilaterals sum to 180°; conversely, any quadrilateral for which this is true can be inscribed in a circle. As another example, the inscribed angle theorem is the basis for several theorems related to the power of a point with respect to a circle. Further, it allows one to prove ...
An inscribed angle (examples are the blue and green angles in the figure) is exactly half the corresponding central angle (red). Hence, all inscribed angles that subtend the same arc (pink) are equal. Angles inscribed on the arc (brown) are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the ...
Alspach's theorem (graph theory) Amitsur–Levitzki theorem (linear algebra) Analyst's traveling salesman theorem (discrete mathematics) Analytic Fredholm theorem (functional analysis) Anderson's theorem (real analysis) Andreotti–Frankel theorem (algebraic geometry) Angle bisector theorem (Euclidean geometry) Ankeny–Artin–Chowla theorem ...
As stated above, Thales's theorem is a special case of the inscribed angle theorem (the proof of which is quite similar to the first proof of Thales's theorem given above): Given three points A, B and C on a circle with center O, the angle ∠ AOC is twice as large as the angle ∠ ABC. A related result to Thales's theorem is the following:
An inscribed figure is not necessarily unique in orientation; this can easily be seen, for example, when the given outer figure is a circle, in which case a rotation of an inscribed figure gives another inscribed figure that is congruent to the original one.
See the figures in this article for examples. The three defining points may also identify angles in geometric figures. For example, the angle with vertex A formed by the rays AB and AC (that is, the half-lines from point A through points B and C) is denoted ∠BAC or ^. Where there is no risk of confusion, the angle may sometimes be referred to ...
Inscribed angle theorem for hyperbolas [10] [11] — For four points = (,), =,,,, ,, (see diagram) the following statement is true: The four points are on a hyperbola with equation y = a x − b + c {\displaystyle y={\tfrac {a}{x-b}}+c} if and only if the angles at P 3 {\displaystyle P_{3}} and P 4 {\displaystyle P_{4}} are equal in the sense ...
The result is a linear system of three equations, which can be solved by Gaussian elimination or Cramer's rule, for example. 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.