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Prove that the circumcenter, orthocentre...

Prove that the circumcenter, orthocentre, incenter, and centroid of the triangle formed by the points `A(-1,11),B(-9,-8),` and `C(15 ,-2)` are collinear, without actually finding any of them.

Text Solution

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The given points are `A(-1,11),B(-9,-8)`, and `C(15,-2)`.
`AB=sqrt(64+361)=sqrt(425)`
`BC=sqrt(576+36)=sqrt(612)`
`AC=sqrt(256+169)=sqrt(425)`
Thus, `AB=ACneBC`.
Therefore, triangle is isosceles, and in an isosceles triangle, all centres are collinear.
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