Cirgle geo problems (1 Viewer)

[Chang]

Ive got 2 circle geometry questions im having problems with, any help is appreciated,

1)
AB, AC are two equal chords of a circle. AP is another chord of the circle which cuts BC at Q. Prove AP.AQ = AB

2)
ABC is an isosceles triangle in which AB = AC and BC>AB. X is a point on CA produced. Y is a point on BC such that angle(BYX)=2angle(YXC). Show that AXBY is a cyclic quadrilateral.

Thx

 

[Rahul]

thought for 1,

AP.AQ = AB^2,

take this as,

AP/AB = AB/AQ

now u can make relationships between those sides for similar triangles. as the ratio of corresponding sides in similar triangles is equal

perhaps i am wrong, just a thought

 

[Affinity]

and for question 2:
find all the angles you can find..
start with angle C

 

[Afro]

Um u sure uve read da question rite for Numba one???
U need one otha fact sumwhere...

 

[Constip8edSkunk]

no you dont, equal angle subtended by a chord in the same sector and u can prove they r similar coz they r equiangular

 

[Afro]

Which angles equal? What sector? huh......?????

 

[BlackJack]

Guys, you could use a step by step to practise your proofs...

Afro:
Q1: (Sake of background) We have a circle, AB, AC, with AP in between them. Q intersection of AP & BC.

Prove similarity of triangles ABP and the ABQ nestled inside:
Triangle ABC is isoceles (AB=AC)
Hence, ^ABC (= ^ABQ) = ^ACB.
Also, ^ACB = ^APB as angles standing on common chord AB on the circumference.

Hence, ^ABQ = ^ APB (1)

The two triangles share angle A. (2)

So the triangles ABQ and APB are equiangular and similar.

Hence by its property, AB/AQ= AP/AB
Multiply to give required ratio.

 

[Chang]

yer thx guys, i got the answer to part 1...

i still cant work out the second one =[

 

[freaking_out]

r there like general tips that u guys have for 3u circular geometry problems?

 

[Affinity]

question 2:
2*YXC = XYB
YXC + XCY = XYB
YXC = XCY
XCY = ABC
ABY = AXY
etc.

3 U cicle geo questions are usually angle bashes hmm think that's about the only thing I could say about it.