True or false? In every triangle, the centroid, incenter, and circumcenter are collinear.
False
Euler's Line passes through the centroid, the circumcentre and the orthocentre of any non-equilateral triangle.
In an equilateral triangle, the centroid, the orthocentre, the circumcentre and the incentre all coincide.
False. In every triangle, the centroid, incenter, and circumcenter do not necessarily lie on the same line.
To understand why, let's briefly define these terms:
1. The centroid of a triangle is the point of intersection of its medians. A median is a line segment connecting a vertex of the triangle to the midpoint of the opposite side.
2. The incenter of a triangle is the center of the circle inscribed within the triangle. It is equidistant from all three sides of the triangle.
3. The circumcenter of a triangle is the center of the circle circumscribed around the triangle. It is equidistant from the three vertices of the triangle.
To check if the centroid, incenter, and circumcenter are collinear in every triangle, we can examine different types of triangles. Let's consider two cases:
Case 1: Equilateral Triangle
In an equilateral triangle, all three sides are equal, and all three angles are equal. The centroid, incenter, and circumcenter of an equilateral triangle coincide at a single point. Therefore, in an equilateral triangle, the centroid, incenter, and circumcenter are collinear.
Case 2: Scalene Triangle
In a scalene triangle, all three sides are of different lengths, and all three angles are also different. For a scalene triangle, the centroid, incenter, and circumcenter do not lie on the same line. They each have their respective positions within the triangle.
Therefore, since there are examples of triangles where the centroid, incenter, and circumcenter are not collinear, the statement is false.