67. A sphere of radius $R$ touches all the lateral faces of a triangular pyramid at the midpoints of the sides of its base. The segment connecting the vertex of the pyramid with the center of the sphere is bisected by the point of intersection with the base of the pyramid. Find the volume of the pyramid.
67. Using the equality of tangents emanating from one point, we will prove that at the base lies a regular triangle and the medians of the lateral faces, drawn to the sides of the base, are equal. From this it will follow that the pyramid is regular.

Answer: $\frac{R^{3} \sqrt{6}}{4}$.