Use the Point tool to draw or construct independent points, points on paths, and points at intersections.
Click in an empty area of your sketch to create an independent point. 

Click a path object — such as a segment, a circle, or the edge of a polygon — to construct a point on the path. When the Point tool is in the right place to construct a point on a path, the path is highlighted, appearing thicker and in a special color. A point constructed on a path can move anywhere along the path, but nowhere else. 

Click the intersection of two path objects — such as a segment and a circle, or a line and a function plot — to construct a point of intersection. When the Point tool is in the right place to construct an intersection, both paths are highlighted, appearing thicker and in a special color. 
In addition to highlighting your targeted objects, Sketchpad displays a message in the status line telling you when you can construct a point on a path or at an intersection.
There are several ways to create points without using the Point tool. For example, clicking with the Arrow tool on an intersection constructs a point of intersection there in the same way the Point tool does. The Compass tool, Straightedge tools, Polygon tools, and most Custom tools may construct their own points as part of their operation. The Construct menu commands Point on Object, Midpoint, Intersection, and other menu commands also construct points.
The color and style of a new point is determined by Sketchpad's Preferences or by the last point for which you set the color or style.