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Circle
Circle
Sector of a Circle
Sector of a Circle
Segment of a Circle
Segment of a Circle
Theorems on Circle
Theorems on Circle
Angle at center of circle
Angle at center of circle
Central Angle - Angle subtended by arc at the center of the circle.
Inscribed Angle - Angle subtended by arc at the circumference of the circle.
The central angle is twice the inscribed angle.
Angle in a semicircle
Angle in a semicircle
The angle in a semicircle is 90°.
Thus, the hypotenuse of a right triangle is the diameter of the circumscribed circle.
This is a special case of the above theorem where the central angle is 180°.
Angles in the same segment
Angles in the same segment
The angles in the same segment are equal.
Opposite angles in cyclic quadrilateral
Opposite angles in cyclic quadrilateral
A cyclic quadrilateral is a quadrilateral which has all its four vertices lying on a circle. It is also sometimes called inscribed quadrilateral.
The opposite angles in a cyclic quadrilateral add up to 180°.
Length of tangents
Length of tangents
The lengths of two tangents from an outside point to a circle are equal, and the angle between the the tangent and the radius is 90°.
Alternate Segments
Alternate Segments
The angle (β) between the tangent and the chord at the point of contact (A) is equal to the angle at point D in the alternate segment.
Intersecting chords theorem
Intersecting chords theorem
The intersecting chords theorem or just the chord theorem describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths of the line segments on each chord are equal.
Intersecting secant theorem
Intersecting secant theorem
The intersecting secant theorem or just secant theorem describes the relation of line segments created by two intersecting secants and the associated circle.
For the secant lines CB and AD that intersect each other at A:
Tangent-secant theorem
Tangent-secant theorem
If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.
For the secant segment CB and tangent segment DA that intersect each other at point A:
Circle and Triangle
Circle and Triangle
Circle Inscribed in a triangle
Circle Inscribed in a triangle
The incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
Circle circumscribed about a triangle
Circle circumscribed about a triangle
The circumscribed circle is the circle that passes through all three vertices of the triangle. The center of the circumscribed circle is the circumcenter of the triangle, the point where the perpendicular bisectors of the sides meet.
Circle circumscribed about a triangle
Circle circumscribed about a triangle
An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.
From the formulas above, the excircles are always larger than the incircle and that the largest excircle is the one tangent to the longest side and the smallest excircle is tangent to the shortest side
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