harder circle geo questions (1 Viewer)

[onebytwo]

from fitzpatrick Ex. 37(a) (this exercise seems ridiculously difficult, i read through most questions and could probably do about 5, these are a few with which i had absolutely no idea)

Q20.
O is point on a given circle. A circle, centre O cuts this given circle at P and Q and a chord PR of the given circle cuts the circle, centre O, in S. Prove that SQ is perpendicular to OR.

Q21. ABCD is a cyclic quadrilateral whose diagonals intersect at E. A circle is drawn through the points A, B and E. prove that the tangent at E to this circle is parallel to CD

Q25.
draw three circles such that each intersects the other two. Prove that the three common chords are concurrrent.

any help with these questions will be appreciated
thanks

 

[modezero]

q20
let angleOSQ = a
then angleOQS = a (radii, isosceles triangle)
so angleSOQ = 180-2a (angle sum of triangle)

similarly
let angleOSP=b
then angleOPS=b
so angleSOP=180-2b

now then, anglePOQ= angleSOQ+angleSOP =360-2a-2b
so anglePRQ=2a+2b-180 (as POQR is cyclic)
but angleRSQ=180-a-b (angle sum on straight line)
so angleRQS= 180-a-b (angle sum of triangle SQR)
thus SQR is isosceles with RS=RQ
and OS=OQ (radii)
as we have 2 pairs of equal sides, then OQRS is a kite
hence SQ is perpendicular to OR

q21
construct the tangent FG such that it is tangent to circle ABE at E. (with F on the AD side of E)
then angleEAB=angleGEB (alternate segment theorem)
but angleEAB=angleBDC (ABCD is cyclic, both angles subtended by BC)
and angleGEB=angleFED (vertically opposite)
so angleFED=angleCDE
as alternate angles are equal, then the tangent is parallel to DC

q25
this is the radical axis theorem

 

[onebytwo]

for the first one how do you know quad. POQR to be cyclic?
also, does anyone have a link to a proof of the radical axis theorem?
thanks for all the help

 

[zeek]

is the radical axis theorem even part of the 4 unit curriculum?...

 

[Affinity]

No, but you can prove it quite easily..

 

[haque]

they in fact asked this question on the radical axis theorem in one of the hsc's

 

[Gecko888]

Another Circle Geometry Problem:

Draw a tangent to a circle, meeting it at N. Draw two chords PN and QN. Draw a line parallel to the tangent such that it meets PN at K and QN at L. Let the perpendicular bisectors of PK and QL meet at a point W. Show that as the line KL varies, the locus of W is a straight line.

What do you think would be the best approach to this?

 

[Wackedupwacko]

draw a diagram... with everything labelled. anyways your question:

After you drew your diagram draw another circle ove the cyclic quad PWQN (opposite interior angles add up to 180). you will find that WN is the diameter of the new circle. Then since WN is perpendicular to the tangent at N it implies that WN is part of the original diameter of the original circle. THEREFORE as KL moves WN moves along the original diameter and thus is a straight line

 

[Gecko888]

lol...and use a ruler too...

Thanks for that. I was having a mental block on that one....