Question and Answers Forum

All Questions      Topic List

Geometry Questions

Previous in All Question      Next in All Question      

Previous in Geometry      Next in Geometry      

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.

$$\mathrm{we}\:\mathrm{had}\:\mathrm{the}\:\mathrm{proof}\:\mathrm{in}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{for} \\ $$$$\mathrm{Q16066},\:\mathrm{16067}\:\mathrm{etc}.,\:\mathrm{indirectly}.\:\mathrm{But} \\ $$$$\mathrm{I}\:\mathrm{can}\:\mathrm{summarise}\:\mathrm{them}\:\mathrm{later}. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com