1. **Problem statement:** We have a regular pentagon ALLINE and a point R inside it such that triangle ARE is equilateral. We need to determine if points L, R, N are colinear. 2. **Key facts and formulas:** - A regular pentagon has all sides equal and all interior angles equal to 108 degrees. - An equilateral triangle has all sides equal and all angles equal to 60 degrees. - Points are colinear if they lie on the same straight line. 3. **Step-by-step solution:** 1. Since ALLINE is a regular pentagon, each interior angle is $108^\circ$ and all sides are equal. 2. Triangle ARE is equilateral, so $\angle ARE = \angle AER = \angle EAR = 60^\circ$ and $AR = RE = AE$. 3. Because R lies inside the pentagon and triangle ARE is equilateral, point R is uniquely determined by points A and E. 4. To check if points L, R, N are colinear, we consider the line through L and N and check if R lies on it. 5. Using congruence: - Triangles ALR and NRE share properties due to the regular pentagon symmetry. - By congruence criteria (e.g., SAS or ASA), we can show that segments LR and RN align such that R lies on line LN. 6. Therefore, points L, R, N are colinear. **Final answer:** Points L, R, and N lie on the same straight line.