The Theorems

On this page, I've set out the seven theorems, so you can check that you drew the right conclusions from the dynamic geometry pages! I've included diagrams which are just dull static geometry, partly as a back-up in case the dynamic pages didn't work on your computer.
If you get "Error. Click for details" where the dynamic geometry ought to be, it may just be worth reloading the page. If that doesn't work, it probably means Geogebra have changed the location of a crucial file, & I haven't updated the pages!!
If so, please let me know.

• First circle theorem - angles at the centre and at the circumference.
• Second circle theorem - angle in a semicircle.
• Third circle theorem - angles in the same segment.
• Fourth circle theorem - angles in a cyclic quadlateral.
• Fifth circle theorem - length of tangents.
• Sixth circle theorem - angle between circle tangent and radius.
• Seventh circle theorem - alternate segment theorem.

Circle Theorem 1

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The angle at the centre is twice the angle at the circumference.

(Note that both angles are facing the same piece of arc, CB)

Circle Theorem 2

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The angle in a semi-cicle is 90°.

(This is a special case of theorem 1, with a centre angle of 180°.)

Circle Theorem 3

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Angles in the same segment are equal.

(The two angles are both in the major segment; I've coloured the minor segment grey)

Circle Theorem 4

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Opposite angles in a cyclic quadrilateral add up to 180°.

 

Circle Theorem 5

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The lengths of the two tangents from a point to a circle are equal.

CD = CE

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Circle Theorem 6

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The angle between a tangent and a radius in a circle is 90°.

 

Circle Theorem 7

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Alternate segment theorem:
The angle (α) between the tangent and the chord at the point of contact (D) is equal to the angle (β) in the alternate segment*.

*Thank you, BBC Bitesize, for providing the precise wording for this theorem!
Here's a link to the their circles revision pages.

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