Legit proof for cyclic quadrilaterals? (1 Viewer)

[nightweaver066]

I was just reading a solution for one of my past papers and they proved a cyclic quadrilateral by showing an arc subtended equal angles.

This was the diagram:

Forgot to include.. TA is a tangent to the circle at A, and AB is a chord. P is a point on AB such that TA = PT. The secant TB cuts the circle at C, and AC and PC are joined.

The question was to prove that TAPC was a cyclic quadrilateral.

I've asked a teacher prior to reading over this solution about this way of proving cyclic quadrilaterals and they said it's not a proper proof.

Can someone clarify for me if this is or not?

Thanks

 

[thoth1]

wtf is that a tangent at C?

 

[nightweaver066]

wtf is that a tangent at C?

Fixed. Secant at C.

 

[funnytomato]

proofs for cyclic quad
1 opposite angles are supplementary
2 exterior angle = interior opposite angle
3 interval subtends equal angles on the same side

http://www.boardofstudies.nsw.edu.au/syllabus_hsc/mathematics-advanced.html#syllabus
page 28 2.9 "If an interval subtends equal angles at two points on the same side of it
then the end points of the interval and the two points are concyclic."

i think there's a neat summary of all the theorems you need to know for the HSC in one of these books written by Jeff Geha

 

[nightweaver066]

proofs for cyclic quad
1 opposite angles are supplementary
2 exterior angle = interior opposite angle
3 interval subtends equal angles on the same side

http://www.boardofstudies.nsw.edu.au/syllabus_hsc/mathematics-advanced.html#syllabus
page 28 2.9 "If an interval subtends equal angles at two points on the same side of it
then the end points of the interval and the two points are concyclic."

i think there's a neat summary of all the theorems you need to know for the HSC in one of these books written by Jeff Geha

Thanks for that, i've always known it was a proof.. Teacher had me confused for a term.

 

[funnytomato]

Thanks for that, i've always known it was a proof.. Teacher had me confused for a term.

when it comes to geometry, it's kinda arbitary which theorems you assume and which ones you'll have to prove
so yeah, that book definitely helps