Determine whether the following statement is true or false. ExplainThe circumcenter of a triangle is always in the interior of the triangle.?

False.

Refer to the diagram below.
The circumcenter of each obtuse triangle is always outside the triangle.

False statement. The circumcenter of a triangle is only inside the triangle if the triangle is acute. Doesn't matter if it's scalene, isosceles, or equilateral. But the circumcenter of an obtuse triangle is outside the triangle, while the circumcenter of a right triangle is on the triangle; it's the midpoint of the hypotenuse.

Now, replace 'circumcenter' with 'incenter', and that's a true statement, because the incenter of a triangle is always inside the triangle.

It's not true. For example, consider a triangle whose three vertices are
(1,0), (1/2,sqrt(3)/2), and (1/2,-sqrt(3)/2). On a unit circle centered at the origin, these three points correspond to angles 0, 60 degrees, and -60 degrees. The circumcenter of the triangle is at (0,0) and does not lie inside the triangle.

https://gyazo.com/17b09b7777906237054709df3df0fd0a