![]() We will now construct an arbitrary isosceles triangle ABC with an altitude through the non-equal side at D as. It need not pass through the opposite vertex of the triangle. A perpendicular bisector is a line perpendicular to a given line, and passing through its centre. (d)The circumcentre is the point of intersection of the perpendicular bisector of each side. An angle bisector is the line joining the vertex and the opposite side such that it divides the angle made at the vertex in two equal halves. (c)The incentre is the point of intersection of the three angle bisectors. ![]() An altitude is the line joining a vertex and the opposite side, such that it is perpendicular to that opposite side. (b)The orthocentre is the point of intersection of the three altitudes. A median is the line joining a vertex to the midpoint of the opposite side. (a)The centroid is the point of intersection of the three medians. We need to find the relationship between the four centres of a triangle, which are the circumcentre, the incentre, the orthocentre and the centroid. If they do, we will check if they coincide or not. We will then check if the circumcentre, the incentre, the orthocentre and the centroid lie on that line or not. We will begin by constructing an isosceles triangle, and make a perpendicular from one vertex to the opposite side, which is the one having a different length from the other two. Hint: The knowledge of different centres of a triangle will be used in this problem.
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