Four centers problem (2)

Geometry Level 3

Let $I$ be the in-center , $H$ be the orthocenter , $G$ be the centroid and $O$ be the circumcenter of a scalene triangle . If the area of $\Delta IHO = 18 \ \text{cm}^2$ , find the value of the area of $\Delta IGO$ in $\text{cm}^2$ .

The answer is 6.

1 solution

Arjen Vreugdenhil
Jan 13, 2018

It is a well-known fact that $HGO$ lie on a straight line, with $HG = 2\:GO$ . This implies that $HO = 3\:GO$ , and thus $A(\triangle IHO) = 3\:A(\triangle IGO)$ . This immediately gives $A(\triangle IGO) = \tfrac13\cdot 18 = \boxed{6}$ .

Four centers problem (2)

The answer is 6.

1 solution

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