1. **Problem Statement:** Given a circle with points E, F, G, H on the circumference and center O, inside the circle there is a triangle with an angle marked 53° at point G. We need to find the measure of angle $\angle EFHI$. 2. **Relevant Formula and Rules:** In a circle, the measure of an inscribed angle is half the measure of the intercepted arc. 3. **Step-by-step Solution:** - Since $\angle G$ inside the triangle is 53°, and points E, F, G, H lie on the circumference, $\angle EFHI$ is related to this angle by the properties of cyclic quadrilaterals and inscribed angles. - The angle $\angle EFHI$ is the angle subtended by the same arc as $\angle G$ but on the opposite side of the circle. - By the inscribed angle theorem, $\angle EFHI = 180^\circ - 53^\circ = 127^\circ$. 4. **Final Answer:** $$\boxed{127^\circ}$$