The sides of a triangle are x+y=1,7y=x and √3y + x=0.Then the following is an interior point of the triangle.

A) circumcentre

B) centroid

C) incentre

D) orthocentre

Ans: B,C how??

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Circumcenter

Circumcenter of the triangle is the center of the circle formed connecting vertices of the triangles.

If the triangle is accute circumcenter is inside the triangle

If it is a right triangle circumcenter is at hypotenuse

If the triangle is obtuse circumcenter is outside triangle.* (By this triangle circumcenter can be omitted from your answer list)*

Centroid

Centroid of a triangle is formed at the point where it medians intercept. It is clear that every median lines are inside the triangle. Therefore centroid also is inside the triangle.*( This is one of your answers.)*

In-center

In-center is the center of the circle formed by circle that is inside the triangle. All legs of the triangles are tangent to this circle. So it is very simple say that in-center is inside the triangle. *(This is another answer)*

Orthocenter

Orthocenter is the point where the altitudes to the triangle meets.

If the triangle is obtuse orthocenter is outside triangle.

*So the correct answer is in-center and centroid.** Answer B and C*

**Sources:**

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