The Triangle Inequality: When Two Sides Fall Short
Isosceles and right triangles, with numbered sides 5, 6, 12 and 11, 12, 23
Here Leonardo tests when three given lengths can close into a triangle. Isosceles and right triangles are shown, a right triangle bisected from its right angle, beside the citations 'proof by the fifth' and 'by the eighteenth'. Triangles are labelled with side numbers 11, 12, 23 and with vertices a, b, c on sides 5, 6, 12, while a row of segments too short to meet illustrates the case. He concludes that such a naming cannot stand in nature, because two joined sides would be less than the third.
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Right triangle bisected from the right angle
Two isosceles triangles and two right triangles divided by a bisector from the right angle carry the citations 'proof by the fifth' and 'by the eighteenth, it is concluded'. The references invoke earlier propositions to justify the division.
A triangle numbered 11, 12, 23
A triangle is labelled with the side numbers 11, 12 and 23, the numerals written in the ordinary (non-mirror) sense. Because 11 and 12 together fall short of 23, the figure cannot in fact close.
Segments too short to form a triangle
On a single base, three pairs of segments are drawn too short to join at an apex. The sketch visualises lengths that cannot be brought together into a triangle.
Sides 5, 6, 12 cannot close
A triangle with vertices a, b, c bears the sides 5, 6 and 12. Leonardo concludes that this way of naming the faces cannot stand in nature, because the two sides joined together would be less than the third.