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Question Number 16739 by Tinkutara last updated on 26/Jun/17

$$\mathrm{Let}M\mathrm{be}\mathrm{a}\mathrm{point}\mathrm{in}\mathrm{the}\mathrm{interior}\mathrm{of}\mathrm{the}$$$$\mathrm{equilateral}\mathrm{triangle}ABC\mathrm{and}\mathrm{let}{A}^{\prime },$$$$B^{\prime }\mathrm{and}{C}^{\prime }\mathrm{be}\mathrm{its}\mathrm{projections}\mathrm{onto}\mathrm{the}$$$$\mathrm{sides}BC,CA\mathrm{and}AB,\mathrm{respectively}.$$$$\mathrm{Prove}\mathrm{that}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{lengths}\mathrm{of}\mathrm{the}$$$$\mathrm{inradii}\mathrm{of}\mathrm{triangles}{MAC}^{\prime },{MBA}^{\prime }\mathrm{and}$$$${MCB}^{\prime }\mathrm{equals}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{lengths}\mathrm{of}\mathrm{the}$$$$\mathrm{inradii}\mathrm{of}\mathrm{trianges}{MAB}^{\prime },{MBC}^{\prime }\mathrm{and}$$$${MCA}^{\prime }.$$

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