matematicas visuales visual math

In this page we are going to see two ways to draw a 15º angle with a ruler and a 'rusty' compass (a compass of fixed opening). These are very simple constructions and there are, of course, many other ways to do the same (bisecting a 60º angle, subtracting 45º from 60º, ...).

They are basic examples of application of the properties of the angles central and inscribed in a circle (although, as we are going to see later, we can use more basic arguments involving some isosceles triangles).

The first construction is straightforward, and the result is very easy to see as a simple case of the property of angles central and inscribed in a circle:

Fifteen degrees angles: how to draw a fifteen degrees angle with ruler and compass. Angles central and inscribed in a circle | matematicasvisuales

Another basic approach follows. In the next image, the triangle CBO is an isosceles triangle, then:

Fifteen degrees angles: using isosceles triangle to justify the construcction | matematicasvisuales

And we can calculate the angle:

Now we are going to see a second construction of a 15º angle:

Again it is an example of the property of the angles central and inscribed in a circle:

Fifteen degrees angles: A second construction of a fifteen degrees angle. Using the central and inscribed angle in a circle | matematicasvisuales

In the next image, we can see that the triangle CB0 is an isosceles triangle, then:

Fifteen degrees angles: Using isosceles triangle to justify the construction | matematicasvisuales

To finish, we can calculate the angle:

MORE LINKS

Central and inscribed angles in a circle | Mostration | Case II
Interactive 'Mostation' of the property of central and inscribed angles in a circle. Case II: When one chord that forms the inscribed angle is a diameter.
Central and inscribed angles in a circle | Mostration | General Case
Interactive 'Mostation' of the property of central and inscribed angles in a circle. The general case is proved.
Wallace-Simson lines
Each point in the circle circunscribed to a triangle give us a line (Wallace-Simson line)
Wallace-Simson lines | Demonstration
Interactive demonstration of the Wallace-Simson line.
Drawing a regular pentagon with ruler and compass
You can draw a regular pentagon given one of its sides constructing the golden ratio with ruler and compass.
Durer's approximation of a Regular Pentagon
In his book 'Underweysung der Messung' Durer draw a non-regular pentagon with ruler and a fixed compass. It is a simple construction and a very good approximation of a regular pentagon.
The Diagonal of a Regular Pentagon and the Golden Ratio
The diagonal of a regular pentagon are in golden ratio to its sides and the point of intersection of two diagonals of a regular pentagon are said to divide each other in the golden ratio or 'in extreme and mean ratio'.