From an outside point, a line cutting an equal segment
Semicircle a b f built so that b c equals a c (prop. 4)
This page opens a new proposition (4) titled "Semicircle, radius, and inserted line." From a point b outside the end a of a line a f, Leonardo draws a second line meeting the first so that the segment from the contact to b equals the segment from the contact to the line's end. He constructs it from two equal intersecting circles centred on a and b, a line d n drawn through their intersections to cut a f at c, and a semicircle a b f centred at c, proving b c equals a c by the definition of the circle. A small semicircle construction with lettered points is drawn at the upper right.
On this page
Point b, line a f, and the equal segments b c and a c
Two equal circles centred on a and b intersect at e and o; the line d n through those intersections is extended to cut a f at c. The line b c is then drawn to the contact at c, and the compass, opened to b c, draws the semicircle a b f centred at c, so that b c equals a c by the definition of the circle.
Marginal statement of the proposition
From a given point outside a line, a second line is to be drawn to the contact of the first such that it cuts off from it a part similar to itself. This is the marginal restatement of the problem worked out in the main column.