John's conclusions are about the properties of perpendicular bisectors in triangles and their importance in geometric constructions. If Eratosthenes had measured a 30-degree angle to the Sun in Alexandria, he would have used proportions to calculate the Earth's circumference, assuming the rays of the Sun are parallel and the Earth is spherical.
The subject of your question pertains to geometry and the concept of perpendicular bisectors in triangles. When constructing perpendicular bisectors in a triangle, each point where the bisectors intersect is equidistant from the vertices of the triangle. This common point of intersection is known as the circumcenter of the triangle, which is the center of the circumscribing circle (circumcircle) around the triangle.
In the specific case of Eratosthenes and his measurement of the Earth's circumference, if he had found that, in Alexandria, the angle to the Sun was 30 degrees, he would have used this information, along with the actual distance between Alexandria and Syene (assumed to be directly south and on the same meridian), to calculate the full circumference of the Earth. His method involved assuming that the Earth was a sphere and that the Sun's rays are parallel, allowing him to use basic proportions to estimate the Earth's circumference based on the angle measured and the distance between the two cities.