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Source: 2019 AMC 8 #24

November 19, 2019
AMC 8

Problem Statement

In triangle ABCABC, point DD divides side AC\overline{AC} so that AD:DC=1:2AD:DC=1:2. Let EE be the midpoint of BD\overline{BD} and let FF be the point of intersection of line BCBC and line AEAE. Given that the area of ABC\triangle ABC is 360360, what is the area of EBF\triangle EBF? [asy] unitsize(1.5cm); pair A,B,C,DD,EE,FF; B = (0,0); C = (3,0); A = (1.2,1.7); DD = (2/3)*A+(1/3)*C; EE = (B+DD)/2; FF = intersectionpoint(B--C,A--A+2*(EE-A)); draw(A--B--C--cycle); draw(A--FF); draw(B--DD);dot(A); label("AA",A,N); dot(B); label("BB", B,SW);dot(C); label("CC",C,SE); dot(DD); label("DD",DD,NE); dot(EE); label("EE",EE,NW); dot(FF); label("FF",FF,S); [/asy] <spanclass=latexbold>(A)</span>24<spanclass=latexbold>(B)</span>30<spanclass=latexbold>(C)</span>32<spanclass=latexbold>(D)</span>36<spanclass=latexbold>(E)</span>40<span class='latex-bold'>(A) </span>24\qquad<span class='latex-bold'>(B) </span>30\qquad<span class='latex-bold'>(C) </span>32\qquad<span class='latex-bold'>(D) </span>36\qquad<span class='latex-bold'>(E) </span>40