Generalisations Gossard perspector

1 generalisations

1.1 generalisation 1
1.2 generalisation 2
1.3 generalisation 3

generalisations

the construction yielding gossard triangle of triangle abc can generalised produce triangles b c   congruent triangle abc , sidelines parallel sidelines of triangle abc.

generalisation 1

this result due christipher zeeman.

let l line parallel euler line of triangle abc. let l intersect sidelines bc, ca, ab of triangle abc @ x, y, z respectively. let b c   triangle formed euler lines of triangles ayz, bzx , cxy. triangle b c   congruent triangle abc , sidelines parallel sidelines of triangle abc.

generalisation 2

paul yiu s generalisation of gossard triangle.

this generalisation due paul yiu.

let p point in plane of triangle abc different centroid g.

let line pg meet sidelines bc, ca , ab @ x, y , z respectively.
let centroids of triangles ayz, bzx , cxy ga, gb , gc respectively.
let pa point such ypa parallel cp , zpa parallel bp.
let pb point such zpb parallel ap , xpb parallel cp.
let pc point such xpc parallel bp , ypc parallel ap.
let b c   triangle formed lines gapa, gbpb , gcpc.

then triangle b c   congruent triangle abc , sides parallel sides of triangle abc.

when p coincides orthocenter h of triangle abc line pg coincides euler line of triangle abc. triangle b c   coincides gossard triangle agbgcg of triangle abc.

generalisation 3

let abc triangle. let h , o 2 points, , let line ho meets bc, ca, ab @ a0, b0, c0 respectively. let ah , ao 2 points such c0ah prallel bh, b0ah prallel ch , c0ao prallel bo, b0ao prallel co. define bh, bo, ch, co cyclically. triangle formed lines ahao, bhbo, chco , triangle abc homothetic , congruent, , homothetic center lies on line oh. if oh line through centroid of triangle abc, problem yiu s generalization of gossard perspector theorem.