MathDB

Octagon

ABCDEFGH

with side lengths

AB = CD = EF = GH = 10

and

BC= DE = FG = HA = 11

is formed by removing four

6-8-10

triangles from the corners of a

23\times 27

rectangle with side

\overline{AH}

on a short side of the rectangle, as shown. Let

J

be the midpoint of

\overline{HA}

, and partition the octagon into

7

triangles by drawing segments

\overline{JB}

\overline{JC}

\overline{JD}

\overline{JE}

\overline{JF}

, and

\overline{JG}

. Find the area of the convex polygon whose vertices are the centroids of these

7

triangles.
[asy] unitsize(6); pair P = (0, 0), Q = (0, 23), R = (27, 23), SS = (27, 0); pair A = (0, 6), B = (8, 0), C = (19, 0), D = (27, 6), EE = (27, 17), F = (19, 23), G = (8, 23), J = (0, 23/2), H = (0, 17); draw(P--Q--R--SS--cycle); draw(J--B); draw(J--C); draw(J--D); draw(J--EE); draw(J--F); draw(J--G); draw(A--B); draw(H--G); real dark = 0.6; filldraw(A--B--P--cycle, gray(dark)); filldraw(H--G--Q--cycle, gray(dark)); filldraw(F--EE--R--cycle, gray(dark)); filldraw(D--C--SS--cycle, gray(dark)); dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); dot(G); dot(H); dot(J); dot(H); defaultpen(fontsize(10pt)); real r = 1.3; label("

A

", A, W*r); label("

B

", B, S*r); label("

C

", C, S*r); label("

D

", D, E*r); label("

E

", EE, E*r); label("

F

", F, N*r); label("

G

", G, N*r); label("

H

", H, W*r); label("

J

", J, W*r); [/asy]

Problem Statement