Dividing triangles by their midpoints into proportional parts
Midpoint lines make a central triangle a fourth of the whole; notes on right angles
This page collects geometric propositions on the proportional subdivision of triangles. Leonardo shows that joining the midpoints of a triangle's three sides produces a central triangle similar to the whole and equal to one fourth of its area, and states that a right triangle's parts always stay in proportion to the whole however it is cut. A tall isosceles triangle, an equilateral triangle, and right triangles carry the constructions, and a faint circle with an inscribed star is also sketched. A final note treats the equal acute angles of a right-angled triangle.
On this page
The midpoint triangle equals one fourth of the whole
For a triangle with two or three equal angles, drawing a line from the midpoint of each side to the midpoint of each other side yields a triangle similar in proportion to the larger one. This inner triangle contains exactly the fourth part of the whole.
Cutting from the midpoint of base to midpoint of side (b, a, c)
Cutting any triangle with two equal angles from the middle of its base to the middle of a side gives a smaller piece equal to one fourth of the whole, namely a b c. A descending series of isosceles triangles, the middle one equilateral, illustrates the rule.
Parts of a divided right triangle stay in proportion
For any angle divided into whatever part, the remainder of the triangle is always in proportion to the whole. A right triangle with base longer than its height is split by two perpendiculars into proportional pieces.
Equal acute angles of a right-angled triangle
When the two acute angles of an orthogonal (right-angled) triangle are equidistant from the right angle, together they equal the right angle. A small isosceles right triangle divided into two halves demonstrates the case.