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Question

Prove that the circumcenter, orthocenter, 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.

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Solution

Distance of AB, BC and CA
AB=(8)2+(19)2AB=64+361=425unitBC=(15+9)2+(6)2=(24)2+36=612unitandAC=(16)2+(13)2=256+109=425unit
ΔABC are Isosceles triangle
Hence, centroid, orthocentre, incentre and circumcentre lie on same line.

1211030_1310474_ans_5e25d8a9408a4ee38f33a666f04b1296.PNG

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