1. what are the properties of the circumcenter of a triangle? . 2. what are the properties of the incenter of a triangle? . 3. the lengths of three sides of a triangle are: n + 3, n + 6, and 3n – 6. find the range of the possible lengths for n.

#1) The circumcenter may lie inside or outside the circle; all of the vertices of the triangle are equally distant from the circumcenter; the circumcenter is the center of the circumcircle.
#2) The incenter is equally distant from all of the sides of the triangle; the incircle around the incenter will touch all sides of the triangle; the biggest circle that can be drawn in a triangle is the incircle.
#3) The range of the possible lengths for n is 3<n<15.

Explanation:
#1 and 2 are properties of the terms, there is no explanation.

#3: The sum of any two sides of a triangle must be larger than the other side. This gives us the first inequality, (n+3)+(n+6)>3n-6.

Combine like terms:
2n+9>3n-6.

Subtract 2n from both sides:
2n+9-2n>3n-6-2n
9>n-6.

Add 6 to both sides:
9+6>n-6+6
15>n.
 
This gives us n<15.

The second inequality we have is (n+6)+(3n-6)>n+3.

Combine like terms:
4n>n+3.

Subtract n from both sides:
4n-n>n+3-n
3n>3.

Divide both sides by 3:
3n/3>3/3
n>1.

The third inequality we have is:
(n+3)+(3n-6)>n+6.

Combine like terms:
4n-3>n+6.

Subtract n from both sides:
4n-3-n>n+6-n
3n-3>6.

Add 3 to both sides:
3n-3+6>6+3
3n>9.

Divide both sides by 3:
3n/3>9/3
n>3.

We have the inequalities n<15, n>1 and n>3. If n is larger than 3, it automatically fits n>1, so we will narrow this to n<15 and n>3. This gives us the range 3<n<15.