harder GCSE vector questions

as far as I am aware these sorts of questions have only been posed by the Edexcel exam board in England (apologies if I've missed some from other boards)

they continue an erstwhile tradition - a general result seems to have been known to and used by Menelaus (~100CE, the logical one rather than the mythological one (maths rather than myths)) in his work on spherical geometry (but not involving vectors, obviously, because they were not invented until late in the 19th century)

a good focus on using vectors to prove (decent) results that can involve many variations of ratios

a powerpoint is here

combined enlargements

what happens (in general) if you do one enlargement scale factor 2 from one centre and then enlarge the resultant shape from another centre, also scale factor 2?

the powerpoint is here
it needs to be downloaded for the animations to work

enlarge the brown triangle, scale factor 2, to obtain the blue triangle

then enlarge the blue triangle from a different centre, to obtain the red triangle

the two enlargements, one after the other

students could either do their own or use these resources

the third triangle can also be obtained from the first via an enlargement

what will the scale factor be and where will the centre of enlargement be?

it seems as if the three centres have some relationship

check this with the ones drawn above

what happens with different scale factors?

the proof of the relationship can be done using vectors

this was done by James Pearce, of Mathspad fame

many thanks for the very neat vector proof, which he did generalise