Question

Let ∆ABC be a triangle with circumcenter O, centroid G, and orthocenter H. Let ϕ be a similarity. Show that the triangle ϕ(∆ABC) = ∆ϕ(A)ϕ(B)ϕ(C) has...

(a) circumcenter ϕ(O). (b) centroid ϕ(G). (c) orthocenter ϕ(H)

Answer 1

PAGE NO. DATE: We will use following 2 observations. $f 1ABLE ICDL then 10CA) $C8)] = 1 6ce) • (0)] if AB LCD then (A)*(B) 1 °CC) POD) if p is a point, PEAB → CP) 6 (A) $($) @ ao out o circumcentre = OA=0B = OC. => 0(0) $CA) = 0() *CB) - 06)CC) = $(0) is the circumcentre of $CABC) suppose A, 8, c' are midpoints of BC, CA, AB. respectively BAT AC Ø (B) & (A') = 4 (A) • (AC). Now; G lies on the line At 3 0 (6) € $ (A) & (A) Similarly $(67€ () $68'), (6) € OCC)CC) $CG) is the centroid. . @ suppose A is the feet of perpendicular from A to BC. => AA, IBC → 0(A) (A) 1 $(8)+(c HEAA, => 0 CH) € °CA) • (A) Similar relation holds for other sides & $CH) is the orthocentre.