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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 circlesintersect, the line

 joining their centres bisects their commonchord at right angles.

centres of touchingcircles

2. Equal arcs on circlesof equal radii subtendequal angles at thecentre, andconversely.

equal arcs, equalangles

3. Equal angles at thecentre stand on equalchords, andconversely.

equal chords, equalanglesORangles standing onequal chordsORangles standing onequal arcs

 AAMT — TOP DRAWER TEACHERS© 2013 Education Services Australia Ltd, except where indicated otherwise. This document maybe used, reproduced, published, communicated and adapted free of charge for non-commercialeducational purposes provided all acknowledgements associated with the material are retained.

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http://topdrawer.aamt.edu.au/Geometric-reasoning/Big-ideas/Circle-geometry/Angle-and-chord-propertieshttp://topdrawer.aamt.edu.au/Geometric-reasoning/Big-ideas/Circle-geometry/Angle-and-chord-propertieshttp://topdrawer.aamt.edu.au/http://topdrawer.aamt.edu.au/Geometric-reasoning/Big-ideas/Circle-geometry/Angle-and-chord-properties

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Theorem Suggested abbreviation Diagram

18. The products of theintercepts of twointersecting secants toa circle from anexternal point.

intersecting secants

 AP !  BP  = CP !  DP

19. Tangents to a circlefrom an external pointare equal.

tangents fromexternal point

20. The angle between atangent and a chord

through the point ofcontact is equal to theangle in the alternatesegment.

angle in alternatesegment

ORangle between tangentand chord

21. The square of thelength of the tangentfrom an external point

is equal to the productof the intercepts of thesecant passingthrough this point.

square of the tangentORintersecting tangent

and secantORtangent and secant

 PT 2 = AP !  PB

 AAMT — TOP DRAWER TEACHERSpage 5 of 6

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Supplementary theorems

Theorem Suggested abbreviation Diagram

1. Two circles touch if

they have a commontangent at the point ofcontact.

tangent of touching

circles

2. If an interval subtendsequal angles at twopoints on the sameside of it then theendpoints of theinterval and the fourpoints are concyclic.

converse of angles inthe same segment

 AAMT — TOP DRAWER TEACHERSpage 6 of 6