MathLabs

Problem 2

Let OO be the circumcentre and HH the orthocentre of an acute triangle ABCABC. Prove that the area of one of △AOH\triangle AOH, △BOH\triangle BOH, and △COH\triangle COH equals the sum of the areas of the other two.
Step 3 of 4: Use the centroid ratio
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AG=2GM  ⟹  d(A,OH)=2d(M,OH)AG=2GM\implies d(A,OH)=2d(M,OH)
Detailed analysis

The centroid divides the median in the ratio AG:GM=2:1AG:GM=2:1. Since GG lies on OHOH, distances to OHOH satisfy d(A,OH)=2d(M,OH)d(A,OH)=2d(M,OH).