# Understand the problem

Let be the circumcenter and be the centroid of a triangle . If and are the circumcenter and incenter of the triangle, respectively,

prove that

prove that

##### Source of the problem

Balkan MO 1996

##### Topic

Geometry

##### Difficulty Level

Easy

##### Comments

Let be the incentre. Euler’s theorem says that . Hence the result actually proves that .

# Start with hints

Do you really need a hint? Try it first!

The distance is easily computable from standard formulae. For example, one can use complex numbers by assuming that and .

The centroid is given by . Hence .

Show that .

Combining all the hints, the problem reduces to proving that . This follows from and .

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We could see Gerretsen & Euler inequality usage here.

Also