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Question Number 17886 by b.e.h.i.8.3.417@gmail.com last updated on 11/Jul/17

Commented by b.e.h.i.8.3.417@gmail.com last updated on 12/Jul/17

1)circle radius=R 2)acute angle between chords:AB , CD=ϕ find: AP^2 +BP^2 +CP^2 +DP^2 . (in terms of: R and ϕ) trial case: when: ϕ=90^• ⇒AP^2 +BP^2 +CP^2 +DP^2 =4R^2

$$\left.\mathrm{1}\right){circle}\:{radius}={R} \\ $$$$\left.\mathrm{2}\right){acute}\:{angle}\:{between}\:{chords}:{AB}\:,\:{CD}=\varphi \\ $$$${find}:\:{AP}^{\mathrm{2}} +{BP}^{\mathrm{2}} +{CP}^{\mathrm{2}} +{DP}^{\mathrm{2}} . \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\left({in}\:{terms}\:{of}:\:\:\:{R}\:\:\:{and}\:\:\:\varphi\right) \\ $$$${trial}\:{case}:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:{when}:\:\varphi=\mathrm{90}^{\bullet} \\ $$$$\:\:\:\:\:\:\:\:\Rightarrow{AP}^{\mathrm{2}} +{BP}^{\mathrm{2}} +{CP}^{\mathrm{2}} +{DP}^{\mathrm{2}} =\mathrm{4}{R}^{\mathrm{2}} \\ $$

Commented by ajfour last updated on 12/Jul/17

what if the chord lengths have a constant ratio k and acute ∠ between them is φ , with radius R?

$$\mathrm{what}\:\mathrm{if}\:\mathrm{the}\:\mathrm{chord}\:\mathrm{lengths}\:\mathrm{have}\:\mathrm{a} \\ $$$$\mathrm{constant}\:\mathrm{ratio}\:\boldsymbol{\mathrm{k}}\:\mathrm{and}\:\mathrm{acute}\:\angle\: \\ $$$$\mathrm{between}\:\mathrm{them}\:\mathrm{is}\:\phi\:,\:\mathrm{with}\:\mathrm{radius}\:\mathrm{R}? \\ $$

Commented by mrW1 last updated on 12/Jul/17

Commented by mrW1 last updated on 12/Jul/17

AP^2 +BP^2 +CP^2 +DP^2 is not constant for a given ϕ, so you can not express it in terms of R and ϕ. the case ϕ=90° is an exception. See examples:

$${AP}^{\mathrm{2}} +{BP}^{\mathrm{2}} +{CP}^{\mathrm{2}} +{DP}^{\mathrm{2}} \:\mathrm{is}\:\mathrm{not}\:\mathrm{constant} \\ $$$$\mathrm{for}\:\mathrm{a}\:\mathrm{given}\:\varphi,\:\mathrm{so}\:\mathrm{you}\:\mathrm{can}\:\mathrm{not}\:\mathrm{express}\:\mathrm{it} \\ $$$$\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{R}\:\mathrm{and}\:\varphi.\:\mathrm{the}\:\mathrm{case}\:\varphi=\mathrm{90}°\:\mathrm{is}\:\mathrm{an} \\ $$$$\mathrm{exception}. \\ $$$$\mathrm{See}\:\mathrm{examples}: \\ $$

Commented by ajfour last updated on 12/Jul/17

i meant AB=k(CD) .

$$\mathrm{i}\:\mathrm{meant}\:\mathrm{AB}=\boldsymbol{\mathrm{k}}\left(\mathrm{CD}\right)\:. \\ $$

Commented by b.e.h.i.8.3.417@gmail.com last updated on 12/Jul/17

mr Ajfour!have you any solution for this case?

$${mr}\:{Ajfour}!{have}\:{you}\:{any}\:{solution}\:{for}\:{this}\:{case}? \\ $$

Commented by ajfour last updated on 13/Jul/17

i think even in this question will not fetch unique answers, given φ,R, and k. Both chord lengths and φ and R are required.See Q.17963..

$$\mathrm{i}\:\mathrm{think}\:\mathrm{even}\:\mathrm{in}\:\mathrm{this}\:\mathrm{question}\:\mathrm{will}\:\mathrm{not} \\ $$$$\mathrm{fetch}\:\mathrm{unique}\:\mathrm{answers},\:\mathrm{given}\:\phi,\mathrm{R}, \\ $$$$\mathrm{and}\:\boldsymbol{\mathrm{k}}.\:\mathrm{Both}\:\mathrm{chord}\:\mathrm{lengths}\:\mathrm{and}\:\phi \\ $$$$\mathrm{and}\:\mathrm{R}\:\mathrm{are}\:\mathrm{required}.\mathrm{See}\:\mathrm{Q}.\mathrm{17963}.. \\ $$

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