Circle Geometry (1 Viewer)

[Kurosaki]

Hello again BOS denizens, I'd like to enquire as to validity of my proof. In the diagram shown, the angles marked in red are equal.
I want to prove that the quadrilateral ABCD is cyclic.
So what I did was produce a circle around triangle ABC, since all triangles are cyclic. Then I said, since angle BAC= angle BDC, then clearly angle BDC is an angle subtended by the same arc BC, hence it must also lie on the circle. Therefore, cyclic quadrilateral?
I'm not sure as to the expression and the validity of the proof however .

 

[Carrotsticks]

Hello again BOS denizens, I'd like to enquire as to validity of my proof. In the diagram shown, the angles marked in red are equal.
I want to prove that the quadrilateral ABCD is cyclic.
So what I did was produce a circle around triangle ABC, since all triangles are cyclic. Then I said, since angle BAC= angle BDC, then clearly angle BDC is an angle subtended by the same arc BC, hence it must also lie on the circle. Therefore, cyclic quadrilateral?
I'm not sure as to the expression and the validity of the proof however .

This is valid, since the converse of that property (angles subtended from the same arc are equal) is also true.

 

[Kurosaki]

Ah OK, thanks heaps Carrotsticks <3.
Also, how come for a similar question in the BOS 2009 HSC exam, they simply showed the two angles as equal and then stated it was a cyclic quadrilateral without going through my proof? Question 5 a (ii)

http://www.boardofstudies.nsw.edu.a...doc/2009-hsc-exam-mathematics-extension-2.pdf
http://www.boardofstudies.nsw.edu.a..._doc/mathematics-ext2-sample-answers-2009.pdf

Isn't that a little...preemptive?

 

[QZP]

Ah OK, thanks heaps Carrotsticks <3.
Also, how come for a similar question in the BOS 2009 HSC exam, they simply showed the two angles as equal and then stated it was a cyclic quadrilateral without going through my proof? Question 5 a (ii)

http://www.boardofstudies.nsw.edu.a...doc/2009-hsc-exam-mathematics-extension-2.pdf
http://www.boardofstudies.nsw.edu.a..._doc/mathematics-ext2-sample-answers-2009.pdf

Isn't that a little...preemptive?

Nope, your one is just long (even redundant). It's just what carrot said: angles subtended from same arc are equal. This only applies to circles so therefore ABCD must be cyclic. Also... IT IS a "circle" geo thereom so...

 

[hit patel]

Hello Kurosaki,
Your method is long to get to the same point. The theorems are provided so the converse can be assumed true and be used for proofs straight away without having to prove them.