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Question Number 64203 by Prithwish sen last updated on 15/Jul/19

$$\mathrm{M}\mathrm{is}\mathrm{the}\mathrm{midpoint}\mathrm{of}\mathrm{segment}\mathrm{AB}.\mathrm{Prove}$$$$\mathrm{that},\mathrm{for}\mathrm{every}\mathrm{point}\mathrm{P}\mathrm{in}\mathrm{space},$$$$\mid \mathrm{PM}\mid ⩽\frac{\mid \mathrm{PA}\mid +\mid \mathrm{PB}\mid }{2}$$

Commented by Prithwish sen last updated on 16/Jul/19

$$\mathrm{Sir},{\mathrm{isn}}^{\prime }\mathrm{t}\mathrm{PB}\mathrm{or}\mathrm{PA}\mathrm{will}\mathrm{be}\mathrm{the}\mathrm{diagonl}\mathrm{if}$$$$\mathrm{triangle}\mathrm{PAB}\mathrm{transform}\mathrm{into}\mathrm{a}\mathrm{parallalogram}.$$

Commented by MJS last updated on 16/Jul/19

$$\mathrm{draw}AB\mathrm{and}M$$$$\mathrm{now}\mathrm{choose}\mathrm{a}\mathrm{point}P\mathrm{and}\mathrm{its}\mathrm{mirror}{P}^{\prime }\mathrm{so}\mathrm{that}$$$$M\mathrm{is}\mathrm{also}\mathrm{the}\mathrm{mid}\mathrm{of}{PP}^{\prime }$$$$\Rightarrow$$$$AB\mathrm{is}\mathrm{one}\mathrm{diagonal}\mathrm{and}{PP}^{\prime }\mathrm{is}\mathrm{the}\mathrm{other}\mathrm{one}$$$$\mid {PP}^{\prime }\mid =2\mid PM\mid ⩽\mid PA\mid +\mid PB\mid$$

Commented by Prithwish sen last updated on 16/Jul/19

$$\because {\mathrm{PAP}}^{\prime }\mathrm{is}\mathrm{a}\mathrm{triangle}.\mathrm{Ok}\mathrm{sir}\mathrm{thank}\mathrm{you}.$$

Commented by MJS last updated on 16/Jul/19

$${\mathrm{you}}^{\prime }\mathrm{re}\mathrm{welcome}$$$$\mathrm{sorry}\mathrm{I}\mathrm{have}\mathrm{no}\mathrm{possibility}\mathrm{to}\mathrm{add}\mathrm{drawings}$$$$\mathrm{at}\mathrm{the}\mathrm{moment}...$$

Commented by Prithwish sen last updated on 16/Jul/19

$$\mathrm{No}\mathrm{sir}\mathrm{you}\mathrm{have}\mathrm{already}\mathrm{done}\mathrm{a}\mathrm{lot}.\mathrm{I}\mathrm{appreciate}$$$$\mathrm{your}\mathrm{effort}\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{again}.$$

Answered by MJS last updated on 16/Jul/19

$$\mathrm{think}\mathrm{of}\mathrm{a}\mathrm{triangle}$$$$\mathrm{sides}AB,PA,PB$$$$\mathrm{now}\mathrm{make}\mathrm{it}\mathrm{a}\mathrm{parallelogram}$$$$\mathrm{obviously}2\mid PM\mid \mathrm{which}\mathrm{is}\mathrm{one}\mathrm{diagonal}$$$$\mathrm{must}\mathrm{be}\mathrm{shorter}\mathrm{than}\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{two}\mathrm{sides}$$

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