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
Answer: is 63.
PRMO, 2018, Question 21
Geometry Vol I to IV by Hall and Stevens
Try with Hints
AB=2DE in triangle \(HG_1G_2\) and triangle \(HDE\) \(\frac{AG_1}{HD}=\frac{G_1G_2}{DE}=\frac{2}{3}\) then \(G_1G_2=\frac{2DE}{3}=\frac{2AB}{3 \times 2}=\frac{AB}{3}\)
triangle \(G_1G_2G_3\) is similar triangle ABC then \(\frac{AreaatriangleABC}{Area G_1G_2G_3}=\frac{AB^{2}}{G_1G_2^{2}}=9\)
then area triangle ABC=\(9 \times area triangle G_1G_2G_3\)=(9)(7)=63.
Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2009 based on Centroid of Triangle.
Problem - Centroid of Triangle (SMO Entrance)
Let M and N be points on sides AB and AC of triangle ABC respectively. If \(\frac {BM}{MA} + \frac {CN}{NA} = 1\) . Can we show that MN passes through the centroid of ABC?
Key Concepts
2D - Geometry
Triangle
Menelaus's Theorem
Check the Answer
Answer: Yes , we can.
Singapore Mathematics Olympiad
Challenges and Thrills - Pre College Mathematics
Try with Hints
If we got stuck in this problem then we can start this problem by applying Menelaus's Theorem.
It states : if a line intersects \(\triangle ABC\) or extended sides at points D, E, and F, the following statement holds: \(\frac {AD}{BD} \times \frac {BE}{EC} \times \frac {CF}{AF} = 1\)
Again let D is the mid point of AC. As \(\frac {BM}{MA} + \frac {CN}{NA} = 1\) then \(\frac {CN}{NA}<1\) where N lies in the line segment CD.From the picture above we can see g is the intersection point between two lines BD and MN. So if we apply Menelaus's Theorem we get :
