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.
- 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.
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.
Other useful links
- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA
Related Program
Math Olympiad Program… taught by Olympians and researchers
The orthocenter has no role in this problem it can be any intersection of cevians.I think so.
Yeah you are correct.