Two equilateral triangles OPQ and ORS of respective sides a and b.
Case 1: a=b then OP=OQ=OR=OS=a=b=r, i.e. (P,Q,R,S) lie on the same circle, irrespective of triangles' orientation. Hence a^2+a.b+b^2=r^2+r.r+r^2=3.r^2
Case 2: a<>b Assume PQ and RS are not parallel. (P,Q,R,S) on the same circle implies that circle's center I is the intersection of PQ and RS bisectors, which is O by construction. Since OP=a<>b=OR we have a contradiction. Therefore PQ//RS.
PQ and RS share a common bisector as depicted on the following figure: http://bleaug.free.fr/8foxes/8foxes354.png
Since P,J,S belong to circle centered in I we have QPR=QJR=QSR=QIR/2=60°. Hence QR=r.sqrt(3)
PQ//RS implies QOR=120°, therefore in triangle OQR: QR^2=OQ^2+OR^2-2.OQ.OR.cos(QOR) which can be rewritten as: 3.r^2=a^2+b^2+a.b
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to be complete, we should also consider the (sub-)case where PQ and RS lie on the same side w.r.t. O as depicted in: http://bleaug.free.fr/8foxes/8foxes354b.png
If a and b represent positive length values, formula becomes: 3.r^2=a^2+b^2-a.b
If we adopt the convention that (a.b) = scalar product (OQ.OS) including sign, the above formula holds in all cases.
Two equilateral triangles OPQ and ORS of respective sides a and b. then arc(PQ) + arc(RS) = 2 . 60 degree =120 degree but arc(PR) = arc(QS) . Then arc(PR) = arc(QS) = 120 degree. and PR = QS = r.sqrt(3) . In the triangle OPR angle(POR) = 120 degree. Then OP^2 + OR^2 + OP.OR = PR^2 and a^2 + b^2 + a.b = r.sqrt(3) όπερ έδει δείξαι. Philip Pydnaios , from Hellas
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Two equilateral triangles OPQ and ORS of respective sides a and b.
ReplyDeleteCase 1: a=b
then OP=OQ=OR=OS=a=b=r, i.e. (P,Q,R,S) lie on the same circle, irrespective of triangles' orientation.
Hence a^2+a.b+b^2=r^2+r.r+r^2=3.r^2
Case 2: a<>b
Assume PQ and RS are not parallel. (P,Q,R,S) on the same circle implies that circle's center I is the intersection of PQ and RS bisectors, which is O by construction. Since OP=a<>b=OR we have a contradiction. Therefore PQ//RS.
PQ and RS share a common bisector as depicted on the following figure:
http://bleaug.free.fr/8foxes/8foxes354.png
Since P,J,S belong to circle centered in I we have QPR=QJR=QSR=QIR/2=60°. Hence QR=r.sqrt(3)
PQ//RS implies QOR=120°, therefore in triangle OQR:
QR^2=OQ^2+OR^2-2.OQ.OR.cos(QOR)
which can be rewritten as:
3.r^2=a^2+b^2+a.b
bleaug
My dream finally comes through i never believe this will happen to me, i am here to share my testimony how dr Ogbeifun help me to cast death spell on my uncle who killed my parents because of his company, and the matter was taking to court and was not giving justice because i was not having any prove not until i came across this great spell caster online and i explain everything that happened to him and he promise to help me cast the spell within 48hours that i should send my uncle full name and his picture that is going to confess before he die which i did as he commanded. within 2days my uncle started confessing and finally die. am grateful for what dr Ogbeifun did for me and with that i promise to share this testimony to all the viewers around the globe,If you are having similar issues please do contact him,
Deleteyou can contact dr Ogbeifun for any death spell, such as to kill your superior in the office and take his or her place, death spell to kill your father and inherit his wealth ,death spell to kill anyone who have scammed you in the past ,spell for increase in salaries, spell for promotion at the office, spell to get your ex lover back,money spell,if things is not working well in your life then you need to contact him via email ogbefunhearlingtemple@gmail.com or call/whatsapp him via +2348102574680 Do this and thank me
My dream finally comes through i never believe this will happen to me, i am here to share my testimony how dr Ogbeifun help me to cast death spell on my uncle who killed my parents because of his company, and the matter was taking to court and was not giving justice because i was not having any prove not until i came across this great spell caster online and i explain everything that happened to him and he promise to help me cast the spell within 48hours that i should send my uncle full name and his picture that is going to confess before he die which i did as he commanded. within 2days my uncle started confessing and finally die. am grateful for what dr Ogbeifun did for me and with that i promise to share this testimony to all the viewers around the globe,If you are having similar issues please do contact him,
Deleteyou can contact dr Ogbeifun for any death spell, such as to kill your superior in the office and take his or her place, death spell to kill your father and inherit his wealth ,death spell to kill anyone who have scammed you in the past ,spell for increase in salaries, spell for promotion at the office, spell to get your ex lover back,money spell,if things is not working well in your life then you need to contact him via email ogbefunhearlingtemple@gmail.com or call/whatsapp him via +2348102574680 Do this and thank me
to be complete, we should also consider the (sub-)case where PQ and RS lie on the same side w.r.t. O as depicted in:
ReplyDeletehttp://bleaug.free.fr/8foxes/8foxes354b.png
If a and b represent positive length values, formula becomes:
3.r^2=a^2+b^2-a.b
If we adopt the convention that (a.b) = scalar product (OQ.OS) including sign, the above formula holds in all cases.
bleaug
Two equilateral triangles OPQ and ORS of respective sides a and b.
ReplyDeletethen arc(PQ) + arc(RS) = 2 . 60 degree =120 degree
but arc(PR) = arc(QS) .
Then arc(PR) = arc(QS) = 120 degree.
and PR = QS = r.sqrt(3) .
In the triangle OPR angle(POR) = 120 degree.
Then OP^2 + OR^2 + OP.OR = PR^2
and a^2 + b^2 + a.b = r.sqrt(3) όπερ έδει δείξαι.
Philip Pydnaios , from Hellas
My dream finally comes through i never believe this will happen to me, i am here to share my testimony how dr Ogbeifun help me to cast death spell on my uncle who killed my parents because of his company, and the matter was taking to court and was not giving justice because i was not having any prove not until i came across this great spell caster online and i explain everything that happened to him and he promise to help me cast the spell within 48hours that i should send my uncle full name and his picture that is going to confess before he die which i did as he commanded. within 2days my uncle started confessing and finally die. am grateful for what dr Ogbeifun did for me and with that i promise to share this testimony to all the viewers around the globe,If you are having similar issues please do contact him,
ReplyDeleteyou can contact dr Ogbeifun for any death spell, such as to kill your superior in the office and take his or her place, death spell to kill your father and inherit his wealth ,death spell to kill anyone who have scammed you in the past ,spell for increase in salaries, spell for promotion at the office, spell to get your ex lover back,money spell,if things is not working well in your life then you need to contact him via email ogbefunhearlingtemple@gmail.com or call/whatsapp him via +2348102574680 Do this and thank me
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ReplyDeletesend me pdf file with solution on my i.d
ReplyDeleteMy dream finally comes through i never believe this will happen to me, i am here to share my testimony how dr Ogbeifun help me to cast death spell on my uncle who killed my parents because of his company, and the matter was taking to court and was not giving justice because i was not having any prove not until i came across this great spell caster online and i explain everything that happened to him and he promise to help me cast the spell within 48hours that i should send my uncle full name and his picture that is going to confess before he die which i did as he commanded. within 2days my uncle started confessing and finally die. am grateful for what dr Ogbeifun did for me and with that i promise to share this testimony to all the viewers around the globe,If you are having similar issues please do contact him,
ReplyDeleteyou can contact dr Ogbeifun for any death spell, such as to kill your superior in the office and take his or her place, death spell to kill your father and inherit his wealth ,death spell to kill anyone who have scammed you in the past ,spell for increase in salaries, spell for promotion at the office, spell to get your ex lover back,money spell,if things is not working well in your life then you need to contact him via email ogbefunhearlingtemple@gmail.com or call/whatsapp him via +2348102574680 Do this and thank me
Hello everyone my is Gabriel sophia from Canada. My husband and I have been married for over 10 years. We met when I was 18 and he was 21. We’ve been through a lot emotionally together. There were several HUGE fights and painful situations in our marriage, but we always seemed to come out stronger on the other side. Out of the blue my husband just sprung the divorce talk on me, I was totally depressed until I found Dr. Ogbeifun website online and I ordered for a Love spell. You won’t believe my husband called me at the exact time this spell caster finished his spell work in 48hrs. I was totally amazed! He is wonderful and his spells work so fast. His Email: ogbefunhearlingtemple@gmail.com call/whatsapp him via +2348102574680 don't hesitate to contact him
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