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Please-solve-Q-16069-Ask-from-me-the-solution-if-needed-and-please-explain-it-

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Question Number 17158 by Tinkutara last updated on 01/Jul/17
Please solve Q. 16069. Ask from me the solution if needed and please explain it.
$$\mathrm{Please}\:\mathrm{solve}\:\mathrm{Q}.\:\mathrm{16069}.\:\mathrm{Ask}\:\mathrm{from}\:\mathrm{me}\:\mathrm{the} \\ $$$$\mathrm{solution}\:\mathrm{if}\:\mathrm{needed}\:\mathrm{and}\:\mathrm{please}\:\mathrm{explain}\:\mathrm{it}. \\ $$
Commented by mrW1 last updated on 01/Jul/17
one can prove case 1: AB not // to CD and AD not // to BC: the locus of P, whose sum of distances to the sides is constant, is a single fixed point. case 2: AB not // to CD but AD // to BC or AB // to CD but AD not // to BC: the locus of P, whose sum of distances to the sides is constant, is a straight segment. case 3: AB // to CD and AD // to BC the locus of P, whose sum of distances to the sides is constant, is any point in interior of ABCD. case 3 means ABCD is parallelogram.
$$\mathrm{one}\:\mathrm{can}\:\mathrm{prove} \\ $$$$\mathrm{case}\:\mathrm{1}:\:\mathrm{AB}\:\mathrm{not}\://\:\mathrm{to}\:\mathrm{CD}\:\mathrm{and}\:\mathrm{AD}\:\mathrm{not}\://\:\mathrm{to}\:\mathrm{BC}: \\ $$$$\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{P},\:\mathrm{whose}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{distances}\:\:\mathrm{to} \\ $$$$\mathrm{the}\:\mathrm{sides}\:\mathrm{is}\:\mathrm{constant},\:\mathrm{is}\:\mathrm{a}\:\mathrm{single}\:\mathrm{fixed}\:\mathrm{point}. \\ $$$$ \\ $$$$\mathrm{case}\:\mathrm{2}:\:\mathrm{AB}\:\mathrm{not}\://\:\mathrm{to}\:\mathrm{CD}\:\mathrm{but}\:\mathrm{AD}\://\:\mathrm{to}\:\mathrm{BC} \\ $$$$\mathrm{or}\:\mathrm{AB}\://\:\mathrm{to}\:\mathrm{CD}\:\mathrm{but}\:\mathrm{AD}\:\mathrm{not}\://\:\mathrm{to}\:\mathrm{BC}: \\ $$$$\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{P},\:\mathrm{whose}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{distances}\:\:\mathrm{to} \\ $$$$\mathrm{the}\:\mathrm{sides}\:\mathrm{is}\:\mathrm{constant},\:\mathrm{is}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{segment}. \\ $$$$ \\ $$$$\mathrm{case}\:\mathrm{3}:\:\mathrm{AB}\:\://\:\mathrm{to}\:\mathrm{CD}\:\mathrm{and}\:\mathrm{AD}\://\:\mathrm{to}\:\mathrm{BC} \\ $$$$\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{P},\:\mathrm{whose}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{distances}\:\:\mathrm{to} \\ $$$$\mathrm{the}\:\mathrm{sides}\:\mathrm{is}\:\mathrm{constant},\:\mathrm{is}\:\mathrm{any}\:\mathrm{point}\:\mathrm{in} \\ $$$$\mathrm{interior}\:\mathrm{of}\:\mathrm{ABCD}. \\ $$$$ \\ $$$$\mathrm{case}\:\mathrm{3}\:\mathrm{means}\:\mathrm{ABCD}\:\mathrm{is}\:\mathrm{parallelogram}. \\ $$
Commented by mrW1 last updated on 02/Jul/17
they can be proved.
$$\mathrm{they}\:\mathrm{can}\:\mathrm{be}\:\mathrm{proved}. \\ $$
Commented by Tinkutara last updated on 02/Jul/17
Will you prove them? Or tell me the source from where I can see their proofs.
$$\mathrm{Will}\:\mathrm{you}\:\mathrm{prove}\:\mathrm{them}?\:\mathrm{Or}\:\mathrm{tell}\:\mathrm{me}\:\mathrm{the} \\ $$$$\mathrm{source}\:\mathrm{from}\:\mathrm{where}\:\mathrm{I}\:\mathrm{can}\:\mathrm{see}\:\mathrm{their} \\ $$$$\mathrm{proofs}. \\ $$
Commented by mrW1 last updated on 02/Jul/17
we had the proof in the solution for Q16066, 16067 etc., indirectly. But I can summarise them later.
$$\mathrm{we}\:\mathrm{had}\:\mathrm{the}\:\mathrm{proof}\:\mathrm{in}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{for} \\ $$$$\mathrm{Q16066},\:\mathrm{16067}\:\mathrm{etc}.,\:\mathrm{indirectly}.\:\mathrm{But} \\ $$$$\mathrm{I}\:\mathrm{can}\:\mathrm{summarise}\:\mathrm{them}\:\mathrm{later}. \\ $$

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