In the NSW maths syllabus, the line joining the points of contact from an exterior point is called the Chord of Contact. I am trying to prove that the pole obtained from your two-secant construction is indeed the chord of contact. You seem to state this without proof in your video. How can I prove that the pole is the chord of contact using common Circle Geometry theorems?
This is so interesting! At first I found the diagram for the Polar Independence Theorem rather daunting, but then I drew up my own version and it all made perfect sense - and the Exercise seemed immediately obvious!
Really elegant stuff!! Thank you so much for going to the effort of putting this together, much appreciated!!!
HI approxdec, The pole of a line theorem (slide 10) states that if A=bc, then the pole of the line A is b^p c^p where p means perp. In other words, choose two points b,c on the line A, and find the meet of their polars. This is the pole of A.
In the NSW maths syllabus, the line joining the points of contact from an exterior point is called the Chord of Contact. I am trying to prove that the pole obtained from your two-secant construction is indeed the chord of contact. You seem to state this without proof in your video. How can I prove that the pole is the chord of contact using common Circle Geometry theorems?
joncui 5 months ago
This is so interesting! At first I found the diagram for the Polar Independence Theorem rather daunting, but then I drew up my own version and it all made perfect sense - and the Exercise seemed immediately obvious!
Really elegant stuff!! Thank you so much for going to the effort of putting this together, much appreciated!!!
flame0154 5 months ago
Thank this wonderful
absolutemes 5 months ago
Thank you sir.
brangelito 7 months ago
have i missed the method for constructing the pole?
other than using tangents, which is a dark art, constructing
the pole is also tricky. your GSP files do it but how to do it
by hand?
approxdec 7 months ago
HI approxdec, The pole of a line theorem (slide 10) states that if A=bc, then the pole of the line A is b^p c^p where p means perp. In other words, choose two points b,c on the line A, and find the meet of their polars. This is the pole of A.
njwildberger 7 months ago
Doing those excercises really made me familiar with the knowledge and capable of using it more effectively. Thank you, this is fascinating!
EclecticSceptic 10 months ago
Im going to compile all of this crap into lecture notes so that I can teach people about this...
QuantumMaths 10 months ago
Excellent. Thanks.
LeavingCertMaths 10 months ago
Thank you
Waranle 10 months ago