Let ABC be a triangle in which AB AC Let L be the locus of points X inside or on the triangle such t...........

Quantitative Aptitude (CDS) Coordinate Geometry

Let ABC be a triangle in which AB = AC. Let L be the locus of points X inside or on the triangle such that BX = CX. Which of the following statements are correct?

1. L is a straight line passing through A and incentre of $\Delta$ ABC is on L.

2. L is a straight line passing through A and orthocentre of $\Delta$ ABC is a point on L.

3. L is a straight line passing through A and centroid of $\Delta$ ABC is a point on L.

Select the correct answer using the code given below.

1 and 2

2 and 3

1 and 3

1,2 and 3

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Explanation:

In DABC, AB = AC i.e. $\Delta$ ABC is isosceles triangle.

If L is the locus of points X inside or on the triangle, such that BX = CX. Then, L is a straight line passing through A.
As, AX is bisecting the line BC, then it passes through the centroid.
As, AB = AC in $\Delta$ ABC, then AX is perpendicular to BC, hence it passes through the orthocentre.
In $\Delta$ ABC, AB = AC and AX $\perp$ BC. So, AX is angle bisector of $\angle$ A, hence it passes through the incentre.
Hence, all statements are correct.