Which of the following are properties of the circumcenter of a triangle? check all that apply. correct answers explain if possible a. the circumcenter of a triangle is not always inside it. b. the circumcenter is equidistant from each vertex of the triangle. c. the circumcenter is equidistant from each side of the triangle. d. the circumcenter is at the intersection of the perpendicular bisectors of the triangle's sides

 Options A, Option B and D are correct.

Step-by-step explanation:

The Circumcenter of a triangle is the center of the circle that passes through all the vertices of the triangle.

According to option B The circumcenter of a triangle is not always inside it.

It is true because In case of obtuse triangle it falls outside the triangle, also, in case of right angled triangle it occurs on the mid point of hypotenuse. So, it is not necessary that the circumcenter of a triangle is not always inside it.

Thus, Option A is true.

According to option B The circumcenter is equidistant from each vertex of the triangle.

Since, The distance of circumcenter from the vertices of a triangle make the radius of the circle. And, radius can not be changed.

Thus, Option B is true.

According to option D The circumcenter is at the intersection of the perpendicular bisectors of the triangle's sides.

By definition of circumcenter, 'The circumcenter of a triangle is the point at which the three perpendicular bisectors of the sides of the triangle intersect'.

Thus, Option D is also true.

Note: Option C is not necessary for all triangles

The circumcenter of a triangle is the point of intersection of the three perpendicular bisectors from the sides of a triangle.

The true statements are: (a), (b) and (d)

From the definition of the circumcenter, we understand that:

It is the meeting point of the perpendicular bisectors of the sides of the triangle.

This means that (d) is correct

It may or may not be equidistant from the sides of the triangle

This means that (c) is incorrect

It is always equidistant from the vertices of the triangle

This means that (b) is correct

Lastly, it is not always inside the triangle (see attachment for illustration)

The above statement is true for obtuse triangles

This means that (a) is correct

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