Let-ABCD-be-a-convex-quadrilateral-and-M-a-point-in-its-interior-such-that-MAB-MBC-MCD-MDA-Prove-that-one-of-the-diagonals-of-ABCD-passes-through-the-midpoint-of-the-other-diagonal-

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Question Number 16064 by Tinkutara last updated on 21/Jun/17

$$\mathrm{Let}ABCD\mathrm{be}\mathrm{a}\mathrm{convex}\mathrm{quadrilateral}$$$$\mathrm{and}\mathrm{M}\mathrm{a}\mathrm{point}\mathrm{in}\mathrm{its}\mathrm{interior}\mathrm{such}\mathrm{that}$$$$\left[MAB\right]=\left[MBC\right]=\left[MCD\right]=\left[MDA\right].$$$$\mathrm{Prove}\mathrm{that}\mathrm{one}\mathrm{of}\mathrm{the}\mathrm{diagonals}\mathrm{of}$$$$ABCD\mathrm{passes}\mathrm{through}\mathrm{the}\mathrm{midpoint}\mathrm{of}$$$$\mathrm{the}\mathrm{other}\mathrm{diagonal}.$$

Commented by Tinkutara last updated on 20/Jun/17

Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 24/Jun/17

$$Bo\times oM.sin\left(BoM\right)=Do\times oM.sin(180-BoM)\Rightarrow$$$$\Rightarrow Bo=Do\Rightarrow ACpassesfrommidpoint$$$$ofBD.$$

Commented by Tinkutara last updated on 27/Jun/17

$$\mathrm{This}\mathrm{is}\mathrm{incorrect}\mathrm{solution}.\mathrm{Should}\mathrm{I}$$$$\mathrm{post}\mathrm{the}\mathrm{correct}\mathrm{answer}?$$

Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 25/Jun/17

$$\left[BCM\right]=\left[CMD\right]\Rightarrow BM=MD$$$$\left[BMA\right]=\left[DMA\right]\Rightarrow \mathrm{\angle }BMA=\mathrm{\angle }DMA$$$$\Rightarrow \{\begin{array}{l}BM=MD\\ \mathrm{\angle }BMo=\mathrm{\angle }DMo\end{array}\stackrel{oM=oM}{\Rightarrow }\mathrm{\Delta }BMo=\mathrm{\Delta }DMo$$$$\Rightarrow Bo=Do\Rightarrow ACpassestroughmidpoint$$$$ofBD.$$

Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 27/Jun/17

$$yesofcorse.pleasepostit.$$

Commented by Tinkutara last updated on 27/Jun/17

Commented by Tinkutara last updated on 27/Jun/17

$$\mathrm{This}\mathrm{is}\mathrm{my}{\mathrm{book}}^{\prime }\mathrm{s}\mathrm{solution}.\mathrm{Please}$$$$\mathrm{explain}\mathrm{the}\mathrm{last}\mathrm{line}.$$

Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 27/Jun/17

$$weprovethat:Do=Bo,anditis$$$$provesthatACpassestroughmidpoint$$$$ofBD.$$

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