how to prove that circumcenter of a triangle ABC with coordinates (x1, y1), (x2 y2), (x3, y3) is given by (x1sin2A + x2sin2B + x3sin2C)/ (sin2A+sin2B+sin2C), (y1sin2A + y2sin2B + y3sin2C)/ (sin2A+sin2B+sin2C)

BACK

Analytical Geometry> how to prove that circumcenter of a trian...

how to prove that circumcenter of a triangle ABC with coordinates (x₁, y₁), (x₂ y₂), (x₃, y₃) is given by (x₁sin2A + x₂sin2B + x₃sin2C)/ (sin2A+sin2B+sin2C), (y₁sin2A + y₂sin2B + y₃sin2C)/ (sin2A+sin2B+sin2C)

gopal singh Negi , 5 Years ago

Grade 12th pass

2 Answers

Arun

Last Activity: 5 Years ago

Let co-ordinate of vertices of triangleABC (x1, y1) , (x2, y2) and (x3 , y3) respectively.

circumcentre :- point of intersection of perpendicular bisector of sides of traingle is known as circumcentre .

circumcentre (C) =

{(x1sin2A+x2sin2B+x3sin2C)/(sin2A+sin2B+sin2C), (y1sin2A+y2sin2B+y3sin2/(sin2A+sin2B+sin2C)}

Aditya Gupta

Last Activity: 5 Years ago

once again, aruns ans is wrong as he has not provided the proof that u were asking for. The actual proof is quite long and involves lengthy calculations, hence i am providing you the steps of the proof, without calculations:

Let the mid pts of BC, CA, AB be D, E, F. obviously perpendiculars to BC, CA, AB at D, E, F intersect at the circumcentre O (by definition). we ll call the circumradius R.
coordinates of D are ((x2+x3)/2, (y2+y3)/2)
extend DO to intersect AB at P.
by properties of circumcircle, DO= RcosA and OP= RcosC/cosB and PB= a/2cosB and PA= c – a/2cosB
so, PO:OD= cosC:cosAcosB and BP:AP= a/2cosB: c – a/2cosB, or by sine rule, BP:AP= 1: 2cosBsinC/sinA – 1.
by section formula, find the coordinates of P, and thence the reqd coordinates of O by again using section formula.

after all of these calculations, you will get the reqd coordinates. it personally took me half an hour to complete all the calculations. also, make sure not to make any mistakes during the entire prob. however, i would recommend u to simply memorise this formula, since the proof is well beyond the scope of jee. cheers!!

KINDLY APPROVE :))