Circle Geometry :( (1 Viewer)

[Forbidden.]

So far I know in my head

Equal arcs subtend equal angles in the same circle
Opposite sides in a cyclic quadrilateral are supplementary
An angle in a semi-circle is a complementary angle

Cant remember the rest ...


Which students here are very clever that they are able to remember all 17 theorems and proofs for Circle Geometry ..

And how ?

 

[pLuvia]

From the top my head, plus some of the theorems are repeats so technically there are less than 17

The angle on the circumference of a circle is twice the angle in the centre
Angles in the same segment
Opposite sides fof the cyclic quadrilateral are supplementary
The angle in a semi circle is right angled
A tangent to a circle is perpendicular to the radius
Tangents from an external point to a circle are equal in length
The angle between a tangent and a chord of a circle is equal to the angle in the alternate segment
A chord is bisected if the radius is perpendicular to it
If two circles touch at the circumference the line passing through both their centres pass through the point of contact

 

[Forbidden.]

but i wanna know ... how on earth do you remember the proofs and theroems well ?

 

[abcd9146]

a few more that havent been listed, based on what i remember....

an angle in a cyclic quad is equal to its opposite exterior one (just more of the basic one)
chords equal distances from the centre are equal
equal angles at centre stand on equal chords
product of intercepts of two intersceting chords are equal

not sure what else... someone get a textbook...

 

[pLuvia]

Oh more I forgot

Let tangent from the point of contact to a circle to an external point be AB. Considering a chord stretched to the same external point, let the chord be CD and the excess length be BC. Then AB²=CD.BD

but i wanna know ... how on earth do you remember the proofs and theroems well ?

It's just a matter of doing hundreds of questions using all these theorems, just do a lot of questions and it'll stay in your head

 

[SoulSearcher]

but i wanna know ... how on earth do you remember the proofs and theroems well ?

Just do thousands of questions (not litereally of course, but quite a few), and they should eventually be stuck in your head. Alternatively, just stick up these theorems on the walls of your room in aplace where you will read them, and that should help as well.

 

[Trev]

You look at the circle and go hey, how can I bullshit about this picture that's been given to me and get marks by doing it...

 

[airie]

Actually, I think I can safely say that there are more than just 17 theorems concerning circles But don't fret, no one will ever give you a question such as "write down as many circle threorems as you can think of", rather it's a matter of actively applying the theorems to the problems. If the question doesn't seem obvious, I personally would look at it and "work backwards", think about what you would need to find in order to get the answer required, and at each step relate to the theorems that concern such things to prove as equal angles, equal sides etc. while looking at what you already can prove using the information provided (a big and accurate diagram can always help and sometimes suggest unexpected results, but always keep in mind that your diagram can only be one particular case at best). And yes, it does sound cliched but practice always helps

 

[priesty]

yeh, i agree with all the above ppl when they say PRACTISE them. in my half yearly they gave us a circle and asked us to prove the alternate segment theorem.

one thing to look at is the relationships between all the theorems. For example look at all the theorems that deal with Tangents and pick out the differences and similarities. From all the theorems about Arcs and Chords do the same. And likewise for all of the Cyclic Quad and segment/sector theorems, and so on. When you read through a couple of times, and learn how to apply to questions you'll understand the theory behind each theorem and will be able to prove it.

Good Luck

 

[airie]

Oh yes, when you mentioned alternate segment theorem, that reminds me of another thing: Always be sure to cover ALL cases in your proof, making it absolutely clear to the marker that the result that you have proven applies to all cases and therefore it does not matter how your diagram looks like (or they sometimes might deduct marks arguing that your proof is diagram-dependent). The proof for the alternate segment theorem, for example, would require three cases

 

[priesty]

my school is also anal with that. and also they make sure that we write a line for "Construct" and we gotta write down which lines we constructed ourselves on the diagram that they usually give you.
Then after you've explained what u've contructed you fill in the "Proof".

 

[airie]

Yep, all those little things are needed for a rigourous proof...which can be a bit annoying at times

 

[priesty]

yeh unfortunately so... i only got 3/5 for that alternate segment theory proof

if only i had known earlier.

 

[onebytwo]

theres this one question in fitzpatrick i was trying to do today...

AB and AC are equal chords of a circle. AD and BE are parallel chords through A and B respectively. Prove that AE is parallel to CD.

I know that i have to prove angle EAD and angle ADC are equal, but i cant make use of the part of thequestion that AB equals AC

anyone, thanks

 

[SoulSearcher]

theres this one question in fitzpatrick i was trying to do today...

AB and AC are equal chords of a circle. AD and BE are parallel chords through A and B respectively. Prove that AE is parallel to CD.

I know that i have to prove angle EAD and angle ADC are equal, but i cant make use of the part of thequestion that AB equals AC

anyone, thanks

I saw this question earlier in a thread, and gave the answer, so I'm not going to type it out twice. Check the last post of this thread for the answer:
http://community.boredofstudies.org/13/mathematics-extension-1/109227/circle-geo.html