Problem 2

Let

O

be the circumcentre and

H

the orthocentre of an acute triangle

ABC

. Prove that the area of one of

\triangle AOH

\triangle BOH

, and

\triangle COH

equals the sum of the areas of the other two.

Step 1 of 4: Place the centroid on the Euler line

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OH\text{ is the Euler line and }G\in OH

Detailed analysis

If O=H (the equilateral case), all three areas are zero and the claim is immediate. Otherwise OH is a line containing G. If no vertex lies on OH, two vertices can be named B,C on the same side; if one vertex lies on OH, its area is zero and the other two areas are equal by the signed-distance relation. Thus we may use the same-side case below.