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Geometry India Math Olympiad Math Olympiad PRMO

Centroids and Area | PRMO 2018 | Question 21

Try this beautiful problem from the Pre-RMO, 2018 based on Centroids and Area. You may use sequential hints to solve the problem.

Try this beautiful problem from the PRMO, 2018 based on Centroids and Area.

Centroids and Area – PRMO 2018


Let ABC be an acute angled triangle and let H be its orthocentre. Let \(G_1\),\(G_2\) and \(G_3\) be the centroids of the triangles HBC, HCA, HAB. If area of triangle \(G_1G_2G_3\) =7 units, find area of triangle ABC.

Centroids and Area- Figure
  • is 107
  • is 63
  • is 840
  • cannot be determined from the given information

Key Concepts


Orthocentre

Centroids

Similarity

Check the Answer


But try the problem first…

Answer: is 63.

Source
Suggested Reading

PRMO, 2018, Question 21

Geometry Vol I to IV by Hall and Stevens

Try with Hints


First hint

AB=2DE in triangle \(HG_1G_2\) and triangle \(HDE\) \(\frac{AG_1}{HD}=\frac{G_1G_2}{DE}=\frac{2}{3}\) then \(G_1G_2=\frac{2DE}{3}=\frac{2AB}{3 \times 2}=\frac{AB}{3}\)

Second Hint

triangle \(G_1G_2G_3\) is similar triangle ABC then \(\frac{AreaatriangleABC}{Area G_1G_2G_3}=\frac{AB^{2}}{G_1G_2^{2}}=9\)

Final Step

then area triangle ABC=\(9 \times area triangle G_1G_2G_3\)=(9)(7)=63.

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