The greatest quadrilateral in a right triangle
Transforming and stretching figures, with a rule proved by its own reversal
Geometrical studies proving that the greatest quadrilateral fitting inside a right triangle equals half of that triangle, its three corners touching the midpoints of the sides. Leonardo then works on stretching quadrilaterals to given lengths and calculating the resulting reduction in their width, illustrated by small hatched right-triangle diagrams at the upper right. He states that a rule which transforms a surface into another figure and can restore the first figure is 'perfect', comparing it to dividing a number and remultiplying it — 12 divided among 3 and the 4 then remultiplied by 3 gives back 12. Crossed-out phrases and marginal notes surround the demonstration.
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Greatest quadrilateral equals half the right triangle
Let a b c d be the greatest quadrilateral, its three angles a, c, d touching the midpoints of the sides of the right triangle e b f. Drawing the diameter b c divides the triangle into four right triangles like the whole; the quadrilateral contains two of them, so it equals half the right triangle.
A perfect rule can be run in reverse
The proof is made by the converse action: the stretching of the parallelogram is shortened and restored to its first figure. A transformation that can be undone confirms the rule.
Proof by division and remultiplication of numbers
As arithmeticians do, a number divided by another and then remultiplied by that same divisor returns the first number. Dividing 12 by 4 into three parts and remultiplying the 4 by 3 remakes 12.