<prompt> Katika pembetatu ya ABC kulia, iliyoonyeshwa hapa, AB 15 text units, AC 24 text units na pointi D, E, na F ndizo sehemu za kati za overlineAC, overlineAB na overlineBC, mtawalia. Katika vitengo vya mraba, eneo la pembetatu DEF ni nini?asyukubwa 150;kuagiza olympiad;jozi A,B,C,D,E,F;A 0,0; B 15, 0; C 0, 24;E katikatiA--B; D katikatiA--C; F katikatiB--C;choraA--B; kuchoraB--C; kuchoraC--A;kuchoraD--F; kuchoraF--E; kuchoraE--D;leboA, A, S;leboB, B, S;leboC, C, N;leboD, D, W;leboE, E, S;leboF, F, NE;choraalama ya kuliaC, A, B, 20;asy  <response>: Kwa kuwa D, E, na F zote ni sehemu za kati, pembetatu zinazoundwa zinalingana angalia picha: overlineDF cong overlineAE cong overlineEB, kwa sababu mstari kuunganisha midpoints mbili katika pembetatu ni sawa, kwa urefu, hadi nusu ya msingi. Vile vile, overlineDE cong overlineCF cong overlineFB na overlineEF cong overlineAD cong overlineDC. Kutoka kwa miingiliano hii, iliyoonyeshwa kwenye picha hapa chini, pembetatu CDF cong pembetatu DAE cong pembetatu FEB cong pembetatu EFD, na SSS, na kwa hivyo lazima zote ziwe na eneo sawa.asyukubwa 150;kuagiza olympiad;jozi A,B,C,D,E,F;A 0,0; B 15, 0; C 0, 24;E katikatiA--B; D katikatiA--C; F katikatiB--C;choraA--B; kuchoraB--C; kuchoraC--A;kuchoraD--F; kuchoraF--E; kuchoraE--D;leboA, A, S;leboB, B, S;leboC, C, N;leboD, D, W;leboE, E, S;leboF, F, NE;choraalama ya kuliaC, A, B, 20;ongezapathticksA--D,2,0.5,10,25; addpathticksD--C,2,0.5,10,25; ongezapathticksE--F,2,0.5,10,25;ongezavijitiA--E,1,0.5,0,25; ongezavijitiE--B,1,0.5,0,25; addpathticksD--F,1,0.5,0,25;ongezavijitiC--F,3,0.5,10,25; ongeza vijiti F--B,3,0.5,10,25; addpathticksD--E,3,0.5,10,25;asyZaidi ya hayo, tunajua kuwa AB 15 text units, AC 24 text units, kwa hivyo kwa kuwa D na E ni sehemu za kati, rightarrow AD frac15101 2 text units, na AE frac242 text units. Kwa hivyo, eneo la pembetatu DEF ni sawa na eneo la triangle AED frac152 cdot frac242 cdot frac12 frac15 cdot 248 15 cdot 3 45 text units2 