# 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...

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?

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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.

I'm not sure what Da Vinci meant when he wrote the following quote;

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

My best guess at the moment says that the Duke of Milan hired Da Vinci to tutor the ducal children, and math must have been one of the tutorial subjects.

Also, I think we can modify the statement to say,

The circle that touches the angles of an equilateral triangle is triple "a properly drawn" triangle that touches the sides of the same triangle.

If we alow the smaller triangle to rotate, or transform into a right or isosceles triangle, then the smaller triangle of the proper area can be drawn.