Can you find the numerous circle properties in the image? Click on the link to learn more.
- Radius is the same length: \( OA = OC\)
- Angle at center is twice angle at circumference: \( 2 \times \angle ABC = \angle AOC \).
- Inscribed angle theorem: \( \angle ABC = \angle APC \).
- Thales' theorem: \( \angle PAC = 90^ \circ \).
- Alternate segment theorem: \(\angle ABC = \angle ACT \)
- Perpendicular tangent theorem: \( \angle OCT = 90 ^ \circ \).
- Tangents to the circle from a point have the same length: \( TA = TC \).
- Opposite angles in a cyclic quadrilateral: \( \angle ABC + \angle CDA = 180^ \circ \).
Here are additional basic properties that are useful to know:
- Equal arcs subtend equal angles and vice versa.
- Equal angles stand on equal chords and vice versa.
- Equal chords are equidistance from the center and vice versa.
- The perpendicular bisector of a chord passes through the center of the circle
- Any three non-colinear points lie on a unique circle.
- Tangent-chord theorem
- Two secants theorem
- The line connecting intersection points of two circles is perpendicular to the line connecting their centers.