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Inside each isosceles triangle the pair of base angles are equal to each other, and are half of 180° minus the apex angle at the circle's center. Adding up these isosceles base angles yields the theorem, namely that the inscribed angle, ψ , is half the central angle, θ .
The angle between a chord and the tangent at one of its endpoints is equal to one half the angle subtended at the centre of the circle, on the opposite side of the chord (tangent chord angle). If the angle subtended by the chord at the centre is 90°, then ℓ = r √2, where ℓ is the length of the chord, and r is the radius of the circle.
Angle AOB is a central angle. A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one (measured in radians). [1]
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: If AC is a diameter of a circle, then: If B is inside the circle, then ∠ ABC > 90° If B is on the circle, then ∠ ABC = 90° If B is outside the circle, then ∠ ABC < 90°.
The nine-point circle is tangent to the incircle and excircles. In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: [28] [29] The midpoint of each side of the triangle; The foot ...
An angle that is part of a simple polygon is called an interior angle if it lies on the inside of that ... for the angle subtended at the center of a circle, ...
In spherical geometry, a spherical quadrilateral formed from four intersecting greater circles is cyclic if and only if the summations of the opposite angles are equal, i.e., α + γ = β + δ for consecutive angles α, β, γ, δ of the quadrilateral. [30] One direction of this theorem was proved by Anders Johan Lexell in 1782. [31]
The area of a triangle is proportional to the excess of its angle sum over 180°. Two triangles with the same angle sum are equal in area. There is an upper bound for the area of triangles. The composition (product) of two reflections-across-a-great-circle may be considered as a rotation about either of the points of intersection of their axes.