Is there a triangle whose vertices are points with rational coordinates on the unit circle and whose vertex angles are 45, 60 and 75 degrees?

 

 

Start with a unit circle with the triangle constructed inside it:

 

 

 

Select a point on the circle and call the coordinates (x,y).

 

Notice that the other points are rotations of the first, about the origin.

 

 

The goal is to find the coordinates of the rotated points, in terms of x and y.

 

Recall that to rotate any point around the origin counterclockwise, by an angle θ you can ěleft multiplyî by the matrix: .

 

 

Rotating the point ( x , y ) by 90 degrees:

 

 

Rotating the new point by 120 degrees:

 

 

 

So the coordinates of a triangle with vertices on the unit circle and vertex angles of 45ƒ, 60ƒ and 75ƒ are:

 

Call these points A, B, and C.

 

 

 

Since we are looking for all rational coordinates, letís make x and y both rational.

 

 

 

That will make the coordinates of A and B rational.

 

The only way for the x-coordinate of C to be rational would be if x=0 or if x is a rational multiple of .

 

We know that x cannot be a rational multiple of Ýbecause x must be a rational number.

 

So x must be 0.

 

If x is 0, then y will be 1 or -1 in order for this point to be on the unit circle.

 

If x is 0 and y is either 1 or -1, then the coordinates of C will be Ýor , not both rational.

 

 

So there is no possible way to make a triangle on the unit circle with vertex angles of 45ƒ, 60ƒ and 75ƒ and have all vertex coordinates rational numbers.

 

 

 

 

 

Change of initial conditions:

 

We can use our solution to the specific problem to create a solution that works for any set of given angles:

 

 

Is there a triangle whose vertices are points with rational coordinates on the unit circle and whose vertex angles are a, b and c degrees?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A simple substitution for the values of a and c will yield the formulas for the coordinates of B and C in terms of x and y.

 

 

 

 

 

 

 

Other possible extensions:

 

1.                 It is obviously possible to create a triangle on the unit circle with all rational coordinates.Ý But is it possible to do this with a triangle that has all rational degree angles?

 

2.                 Is it possible to make certain quadrilaterals (or other polygons) on the unit circle with all rational vertex coordinates?

 

3.                 Is it possible to make ěrational trianglesî on other conics?Ý Perhaps on parabola or ellipses?

 

4.                 Is it possible to make triangles (or other polygons) with rational areas on unit circles (or other conics)?

 

 

 

 

 

 

Jenn daSilva and Jeff Downin

Spring 2007