Fundamental property to be known

Geometry Level 3

Let ABC be a triangle such that the three medians divide it into six parts of equal area. Then , the triangle

can be any triangle cannot exist must be isosceles must be equilateral

1 solution

Julian Poon
Nov 30, 2014

Consider this image:

As you can see, $a+b+c=f+e+d$ $a+f+e=b+c+d$ $a+b+f=e+c+d$

Solving these you will get

$d=a$ ,

$e=b$ ,

$f=c$

Now refer back to the image, it is clear that

$f=a$ ,

$b=c$ ,

$e=d$

Therefore,

$\boxed{a=b=c=d=e=f}$

how you can so sure that f=a..

Siddharth Nahar - 6 years, 6 months ago

Ok. By cutting a triangle from its point to the center of the line segment opposite of the point, on all $3$ of its points, the intersections of these cuts will be the triangle's geometric centroid. This means that the cuts cut the triangle into half. Applying this to the triangle formed by $a$ and $f$ , it shows that $a=f$ . Alternatively, I can show you an image.

You can see that area $A$ and $B$ is half of the parallelogram. And the line cuts the parallelogram cleanly in half.

btw, your equation (f=a) looks a lot like (f=ma) and for a moment I thought it so. Just saying...

Julian Poon - 6 years, 6 months ago

As the meridian divides side of the triangle at mid point, so base line of "a" and "b" are equal, and height from the base line to the centroid of the triangle is same . Area of triangle is 1/2 x base x height. Hence area of both the triangles are equal.

Altaf Ahmed - 6 years, 6 months ago

https://brilliant.org/problems/find-the-areaonly-your-logic-can-help-you/?group=3UHxOzwinQpA&ref_id=384997 PLEASE TRY TO DO THIS AWSOME PROBLEM TOO..post a solution if you get......................i am waiting for an awesome solution that i made while creating this problem

Yash Sharma - 6 years, 3 months ago

Fundamental property to be known

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