Geometry: locating a point equidistant from two given points
Two circle constructions: the perpendicular bisector and finding a midpoint on an arc.
This page continues Leonardo's elementary geometry, giving the 2nd and 3rd propositions of a series. The first construction places a third point n equidistant from two given points a and b by drawing two equal intersecting circles and the line d-e through their intersections. The second finds the middle between two points b and c lying on an arc, again using the line drawn between the intersections of two circles. Two pen diagrams at the right illustrate the constructions.
On this page
Placing a point equidistant from two given points a and b
Given two points a and b, a third point n is to be set equally distant from each. Leonardo justifies it by the preceding proposition: the line d-e passes through the intersections of the two equal circles, and n lies on that line, so it must be equidistant from a and b.
Finding the midpoint between two points on an arc
To find the middle between b and c on the line a-c, he draws the straight line e-d between the two intersections of circles n and m. Where e-d cuts the curved line b-c, at o, lies the midpoint between b and c, which are the centres of those circles.