Let-M-be-a-point-in-interior-of-ABC-Three-lines-are-drawn-through-M-parallel-to-triangle-s-sides-thereby-producing-three-trapezoids-Suppose-a-diagonal-is-drawn-in-each-trapezoid-in-such-a-way-tha

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Question Number 16951 by Tinkutara last updated on 28/Jun/17

$$\mathrm{Let}M\mathrm{be}\mathrm{a}\mathrm{point}\mathrm{in}\mathrm{interior}\mathrm{of}\mathrm{\Delta }ABC.$$$$\mathrm{Three}\mathrm{lines}\mathrm{are}\mathrm{drawn}\mathrm{through}M,$$$$\mathrm{parallel}\mathrm{to}{\mathrm{triangle}}^{\prime }\mathrm{s}\mathrm{sides},\mathrm{thereby}$$$$\mathrm{producing}\mathrm{three}\mathrm{trapezoids}.\mathrm{Suppose}\mathrm{a}$$$$\mathrm{diagonal}\mathrm{is}\mathrm{drawn}\mathrm{in}\mathrm{each}\mathrm{trapezoid}\mathrm{in}$$$$\mathrm{such}\mathrm{a}\mathrm{way}\mathrm{that}\mathrm{the}\mathrm{diagonals}\mathrm{have}\mathrm{no}$$$$\mathrm{common}\mathrm{endpoints}.\mathrm{These}\mathrm{three}$$$$\mathrm{diagonals}\mathrm{divide}ABC\mathrm{into}\mathrm{seven}$$$$\mathrm{parts},\mathrm{four}\mathrm{of}\mathrm{them}\mathrm{being}\mathrm{triangles}.$$$$\mathrm{Prove}\mathrm{that}\mathrm{the}\mathrm{area}\mathrm{of}\mathrm{one}\mathrm{of}\mathrm{the}\mathrm{four}$$$$\mathrm{triangles}\mathrm{equals}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{the}\mathrm{areas}$$$$\mathrm{of}\mathrm{the}\mathrm{other}\mathrm{three}.$$

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