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# 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.

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#1) The circumcenter can be inside or outside the circle; all the vertices of the triangle are the same distance from the circumcenter; the circumcenter is the center of the circumcircle. #2) The incenter is the same distance from all sides of the triangle, the inscribed circle around the incenter will touch all the sides of the triangle is the largest circle you can draw a triangle is the incircle. #3) The range of possible lengths of n is 33n-6. Combine like terms: 2n+9>3n-6. Subtract 2n from both sides: 2n+9-2n>3n-6-2n9>n-6. Add 6 to both sides: 9+6>n-6+615>n. This gives us that n<15. The second inequality that we have is (n+6)+(3n-6)>n+3. Combine like terms: 4n>n+3. Subtract n from both sides: 4n-n>n+3-n3n>3. Divide both sides by 3:3n/3>3/3n>1. The third inequality that 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-n3n-3>6. Add 3 to both sides: 3n-3+6>6+33n>9. Divide both sides by 3: 3n/3>9/3n>3. We have the inequalities n<15, n>1 and n>3. If n is greater than 3, is automatically set to n>1, so we're going to tighten this to n<15 and n>3. This gives us the range of 3