Problem 9

The construction begins with three lines that intersect to form . Choose a point D on ray  so that  is longer than , , and . Draw a circular arc starting at point D, using the point A as center, until it hits. Label this point E.  Draw another circular arc starting at point E, using the point C as center, until it hits . Label this point F. Continue drawing circular arcs around the figure using the natural point as the center to get the following figure in which  = , = ,  = . Prove  = .

[Problem submitted by Kevin Windsor, LACC Instructor of Mathematics.]

Solution for Problem 9:

By construction the following statements are true:

=  + ,  =  + , and  = è      +  =  + .

=  + ,  =  + , and  =    è      +  =  + .

=  + ,  =  + , and  =       è      +  =  +

Also,                                                                               =

and                                                                                = .

Add the left sides and rights side to form the equation:

+++++++ = +++++++

+++++++ = +++++++

Thus,  = .