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# Centroids and Area | PRMO 2018 | Question 21

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

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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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

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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### 2 comments on “Centroids and Area | PRMO 2018 | Question 21”

1. Rishav says:

The orthocenter has no role in this problem it can be any intersection of cevians.I think so.

1. Srijit Mukherjee says:

Yeah you are correct.