Do you agree with this math statement?

Question

'The circumcentre of a triangle is always inside the triangle' Do you agree? Give reasons for your answer. If you agree state why, and if you disagree state why.

Answer 1

No, it isn't. For example, draw a tall isosceles triangle with a narrow horizontal base, then move the top vertex (the vertex opposite the base) far to the left, to make a significantly obtuse triangle. The circumcentre in this situation is obviously far outside the triangle. You could also show it mathematically by doing the calculations for a triangle of this approximate shape.



*Edit*: As polyhymn pointed out, the circumcentre is never inside the triangle (and is in fact always outside the triangle) for *any* obtuse triangle. The circumcentre for any right triangle is at the midpoint of the hypotenuse, which is on the triangle rather than being inside it. See the Wikipedia link in Sources for details.

Answer 2

The circumcenter of a triangle is NOT always inside the triangle. The way you determine the circumcenter is to first find the midpoints of all three sides, then draw perpendicular lines thru all the midpoints and where these lines intersect is the circumcenter. It can easily be outside on a fairly obtuse triangle. Orthocenters are also NOT always inside the triangle.

Answer 3

A very obtuse triangle may have its circumcenter outside the triangle. Such a description would be provided by an illustration, which this text box does not allow me to do.

Answer 4

The circumcenter of an obtuse triangle is never inside the triangle.

Answer 5

Yes it is true. Because it can not be anywhere else. Take note of the definition of circumcentre.