From The Notebooks of Leonardo da Vinci edited by Edward McCurdy

Manuscript G 17 r

When two circles touch the same square at four points, one is double the other.

Manuscript G 17 v

The circle that touches the angles of an equilateral triangle is triple the triangle that touches the sides of the same triangle.

As near as I can tell, the statemtn from G17r is true, but the statement from G17v is not true.

Am I missing something? Is there any way that the statement from G17v can be true?

Expert Answers

| Certified Educator

G 17 r must be true. If r is the radius of the inscribed circle, then the radius of the square is `rsqrt(2)` which is also the radius of the circumscribed circle.

The area of the inscribed circle is `pir^2` and the area of the circumscribed circle is `pi(rsqrt(2))^2=2pir^2` -- so the area of the circumscribed circle is twice the area of the inscribed circle.

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As stated here, G 17 v is false.

Let the radius of the inscribed circle be r. Then the radius of the equilateral triangle (and of the circumscribed circle) is 2r.

The area of the inscribed circle is `pir^2` and the area of the circumscribed circle is `pi(2r)^2=4pir^2` which is quadruple the area of the inscribed circle. The ratio of the circumferences is 2:1. Neither measure is 3 times that of the other.

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