Question

If G (g), H (h) and C (c) are centroid, orthocentre, and circumcenter of a triangle and $xc + yh + zg = 0$ then (x, y, z) =

• A. 1, 1, -2
• B. 2, 1, -3
• C. 1, 3, -4
• D. 2, 3, -5

Answer 1

Answer: (B)

(2, 1, -3)

Answer 2

It is a well–known fact that the centroid $G$, circumcenter $C$ and orthocentre $H$ are collinear along the Euler line. Moreover, they satisfy

$$H = 3G - 2C.$$

Rearranging we get

$$2C + H - 3G =0,$$

which shows that the coefficients $(x,y,z)$ in

$$x\,C+y\,H+z\,G=0$$

are $(2,\, 1,\, -3)$.

Thus, the answer is (2, 1, -3) which corresponds to option (b).