Consider a circle which cuts an ellipse in precisely four points. Let A, B, C and D respectively correspond to the eccentric angles of these four points.

Show that A + B + C + D is an even multiple of pi radians.

I try to resolve this problem but first of all i must study! 

I found at

http://mathworld.wolfram.com/Ellipse.html

this

Let four points on an ellipse with axes parallel to the coordinate axes have angular coordinates. Such points are concyclic when

I will prove now that this relation is same with

A+B+C+D = 2*k*pi

Divide by 2 i found

A/2+B/2+C/2+D/2 = k*pi

or

A/2+B/2 = k*pi - (C/2+D/2)

using tangent function i found

tan(A/2+B/2) = tan(k*pi-(C/2+D/2))

so

(tan(A/2)+tan(B/2)) / (1-tan(A/2)*tan(B/2)) = - (tan(C/2)+tan(D/2)) / (1-tan(C/2)*tan(D/2))

With the notation

s1=tan(A/2), s2=tan(B/2), s3=tan(C/2), s4=tan(D/2)

(s1+s2)/(1-s1*s2) = -(s3+s4)/(1-s3*s4)

and after calculus i found the above relation (31)

 

 

Edited on September 29, 2007, 12:46 pm

Posted by Chesca Ciprian on 2007-09-29 12:43:37