We want to prove that triangle OAC is equilateral, which means we need to show that OA = AC = OC.
Since angle LABC is given as 30 degrees, we can infer that angle LOA (formed by radii OB and OA) is 60 degrees. This is because the angle at the center of the circle is twice the angle at the circumference (angle LOA = 2 * angle LABC = 2 * 30 = 60 degrees).
In a circle, an angle formed by a chord (AC in this case) and a tangent (OA) at the same endpoint (A) is equal to half the central angle (LOA) that intercepts the same arc (BC). Therefore, angle OAC is equal to half of angle LOA, which is 30 degrees. In an equilateral triangle, all three angles are equal, and therefore, all three sides are equal in length. Since angles OAC and OCA are 30 degrees each, triangle OAC must be equilateral, and thus, OA = AC = OC.by using the property of inscribed angles and the given information about the angle LABC, we have established that triangle OAC is equilateral.