Question

Draw other triangles (compare acute, obtuse, right triangles) and construct the centroid, incenter, circumcenter, and orthocenter of each triangle. Answer each of the following questions and explain how you know.

1. Does the centroid always lie inside the triangle?

2. Does the incenter always lie inside the triangle?

3. Does the circumcenter always lie inside the triangle?

4. Does the orthocenter always lie inside the triangle?

5. Can these points ever coincide?

Answer 1

1. Yes the centroid always lies inside the triangle because it is the intersection point of the medians i.e. the lines joining the vertex to the mid-point of opposite side.

Answer 2

2. Yes, the incenter always lies inside the triangle because it is the intersection point of the angle bisectors which meet inside the triangle only.

Answer 3

3. No, the circumcentre does not always lie inside the triangle. Cicumcentre is the point at which the perpendicular bisectors of the three sides meet. In case of a Right Triangle, the circumcentre lies at the mid-point of the hypotenuse. For certain other trianlges it can even lie outside the triangle.

Answer 4

4. No, the orthocentre does not always lie inside the circle. In case of a Right Triangle, it lies on the vertex where the right angle is. For obtuse angles triangles, it lies outside the triangle.

Answer 5

5. Yes, for an equilateral triangle, all the four coincide and so the same point represents all the four.