Question

if centoid and ortocentre of a triangle are (1,2)and (9,6) then circumcentre is​

Answer 1

Answer:

Step-by-step explanation:

In any triangle, orthocentre, centroid, and circumcentre are collinear and centroid divides the line joining orthocentre and circumcenter in ratio 2:1

let coordinates of circumcentre be(x,y)

by formula of internal division

1=9+2x/3

x=-3

2=6+2y/3

y=0

therefore coordinates of circumcentre=(-3,0)

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Answer 2

The relation between the circumference, the orthocenter, and the centeroid is shown in the attachment.

Where g is centrality,
C is the circumcenter
And oh is the orthocenter.


Now we can kind of solve it the other way,
Try it this way,

G is the point which divides oc into two parts,

(see the second attachment for better understanding the way I did)

Now using the sectional formula,


$(\frac{ m_{1} x_{2} + m_{2} x_{1} }{x_{1} + x_{2} } , \frac{ m_{1} y_{2} + m_{2} y_{1} }{x_{1} + x_{2} } ) = (x ,y)$


Solving this,
for x,


$\frac{2x + 9}{ 3 } = 1 \\ 2x + 9 = 3 \\ 2x = - 6 \\ x = - 3$

Sinilarly for y,

$\frac{2y + 6}{3} = 2 \\ 2y + 6 = 6 \\ 2y = 0 \\ y = 0$



Therefore the coordinates of circumcenter are (-3,0)