Circle Geometry Properties

Can you find the numerous circle properties in the image? Click on the link to learn more.

1. Radius is the same length: \( OA = OC\)
2. Angle at center is twice angle at circumference: \( 2 \times \angle ABC = \angle AOC \).
3. Inscribed angle theorem: \( \angle ABC = \angle APC \).
4. Thales' theorem: \( \angle PAC = 90^ \circ \).
5. Alternate segment theorem: \(\angle ABC = \angle ACT \)
6. Perpendicular tangent theorem: \( \angle OCT = 90 ^ \circ \).
7. Tangents to the circle from a point have the same length: \( TA = TC \).
8. Opposite angles in a cyclic quadrilateral: \( \angle ABC + \angle CDA = 180^ \circ \).

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Here are additional basic properties that are useful to know:

1. Equal arcs subtend equal angles and vice versa.
2. Equal angles stand on equal chords and vice versa.
3. Equal chords are equidistance from the center and vice versa.
4. The perpendicular bisector of a chord passes through the center of the circle
5. Any three non-colinear points lie on a unique circle.
6. Tangent-chord theorem
7. Two secants theorem
8. The line connecting intersection points of two circles is perpendicular to the line connecting their centers.

• Circles
• Cyclic quadrilaterals
• Ptolemy's theorem