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Circle Geometry Theorems Geometry/Angle-And-Chord-Properties Circle geometry theorems http://topdrawer.aamt.edu.au/Geometric-reasoning/Big-ideas/Circle- geometry/Angle-and-chord-properties Theorem Suggested abbreviation Diagram 1. When two circles centres of touching intersect, the line circles joining their centres bisects their common chord at right angles. 2. Equal arcs on circles equal arcs, equal of equal radii subtend angles equal angles at the centre, and conversely. 3. Equal angles at the equal chords, equal centre stand on equal angles chords, and OR conversely. angles standing on equal chords OR angles standing on equal arcs AAMT — TOP DRAWER TEACHERS © 2013 Education Services Australia Ltd, except where indicated otherwise. This document may be used, reproduced, published, communicated and adapted free of charge for non-commercial educational purposes provided all acknowledgements associated with the material are retained. page 1 of 6 Theorem Suggested abbreviation Diagram 4. The angle at the angles at the centre centre is twice the and circumference angle at the circumference subtended by the same arc. 5. The tangent to a circle tangent perpendicular is perpendicular to the to radius radius drawn to the point of contact and conversely. 6. The perpendicular perpendicular from from the centre of a the centre circle to a chord bisects the chord. 7. The line from the line joining centre to centre of a circle to midpoint of chord the midpoint of a chord is perpendicular to the chord. 8. The perpendicular perpendicular bisector of a chord bisector of passes through the chord centre of the circle. AAMT — TOP DRAWER TEACHERS page 2 of 6 Theorem Suggested abbreviation Diagram 9. Equal chords in equal equal chords circles are equidistant equidistant from from the centres. centre 10. Chords in a circle equal chords which are equidistant equidistant from from the centre are centre equal. 11. Any three non- perpendicular collinear points lie on bisector of chord a unique circle, whose passes through the centre is the point of centre concurrency of the perpendicular bisectors of the intervals joining the points. 12. Angles in the same angles in the same segment are equal. segment 13. The angle in a semi- angle in a semi-circle circle is a right angle. AAMT — TOP DRAWER TEACHERS page 3 of 6 Theorem Suggested abbreviation Diagram 14. Opposite angles of a opposite angles in a cyclic quadrilateral cyclic quad are supplementary. x + y = 180 15. The exterior angle at a exterior angle of cyclic vertex of a cyclic quad quadrilateral is equal to the interior opposite angle. 16. If the opposite angles converse of opposite in a quadrilateral are angles in a cyclic quad supplementary then the quadrilateral is cyclic. Note: This theorem is also a test for four points to be concyclic. If x + y = 180 then ABCD is a cyclic quadrilateral. 17. The products of the intersecting chords intercepts of two intersecting chords are equal. AP × BP = CP × DP AAMT — TOP DRAWER TEACHERS page 4 of 6 Theorem Suggested abbreviation Diagram 18. The products of the intersecting secants intercepts of two intersecting secants to a circle from an external point. AP × BP = CP × DP 19. Tangents to a circle tangents from from an external point external point are equal. 20. The angle between a angle in alternate tangent and a chord segment through the point of OR contact is equal to the angle between tangent angle in the alternate and chord segment. 21. The square of the square of the tangent length of the tangent OR from an external point intersecting tangent is equal to the product and secant of the intercepts of the OR secant passing tangent and secant through this point. PT2 = AP × PB AAMT — TOP DRAWER TEACHERS page 5 of 6 Supplementary theorems Theorem Suggested abbreviation Diagram 1. Two circles touch if tangent of touching they have a common circles tangent at the point of contact. 2. If an interval subtends converse of angles in equal angles at two the same segment points on the same side of it then the endpoints of the interval and the four points are concyclic. AAMT — TOP DRAWER TEACHERS page 6 of 6 .
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