Geometry: proportional parts of a triangle cut parallel to its base
A single cut divides each nested rectilinear triangle in the same proportion
Continuing the study of the previous folio, Leonardo restates that any cut made parallel to the base of a rectilinear triangle is smaller than that base by the amount the base exceeds it. He then proves that the parts of such a cut stand to the parts of the base in the same ratio that the whole cut has to the whole base, using triangle a b c cut by the line d e. A marginal note observes that the same line bisects every rectilinear triangle nested within the largest one, and that as one side grows above the cut another diminishes below it. The inverted triangle at upper right carries the labels a n m o b and d L r e converging at c.
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Each cut parallel to the base is smaller than the base
Leonardo states the governing conception: each cut made parallel to the base of a rectilinear triangle is smaller than the base by exactly the amount by which the base exceeds the cut. The figure carries the labels a n m o b across the base and d L r e across the cut, meeting at apex c.
Parts of the cut proportional to parts of the base
The parts of the cut made parallel to the base stand to the corresponding parts of the base in the same ratio that the whole cut has to the whole base. He proves it by showing that triangle a b c, cut parallel to base a b by the cut d e, is divided into unequal parts by the straight lines descending from the base to the opposite angle.
One line bisects every nested triangle
The marginal note argues that the line d e, placed through the middle of triangle a b c, likewise bisects triangle o b c at r e and every rectilinear triangle that can be drawn within the largest one. As much as side o b grows above r e, side L e diminishes below side m o.