Locus of a point

                                By Na Young

 

8. Of a triangle, given two vertices A and B, and the angle

at the third vertex C (the angle opposite side AB). What is the locus of the point C?

 

  LetÕs investigate the special case.

  Given two vertices A and B and assume the angle at the third vertex

 C is a right angle. We know that a triangle which one side is a  

 diameter of the CIRCUMCIRCLE has a right angle.

         

            Picture

 

                 

 

 If any point on the circle is the third point, the angle at the third

vertex is 90 degrees. From this, we can think a similar situation.

For an example, make the angle at the third vertex C is 51 degrees.

 

             Picture

                          

  Using GSP choose a mid point M of given points A and B and draw

 a perpendicular line from M. We draw three perpendicular lines from

 midpoints of three sides. Then we can obtain a CIRCUMCENTER of

 a triangle ABC, We call the CIRCUMCENTER O.

 

            Picture

 

                                   

Now we will investigate the change of an angle C.

Move the point C on the circle O.

 

Picture

                      

 

 

   On the above circle of the segment AB the angle C is not changed

 and on the below of the segment AB the angle C has a different

 value of a given angle: That is, in upper part of the segment AB

 (containing a point C) the angle ABC is always 51 degrees.

  The locus of the point C is a part of the CIRCUMCIRCLE of a

  triangle ABC. Explicitly the locus is an arc ACB.

 

    Another interest point is that the angle C is not changed on the

   lower part of the segment AB.

 

     Generally, we can say when given two vertices A and B and the

   angle at the third vertex C, the locus of the point C is a part of the

   CIRCUMCIRCLE of a triangle ABC: Explicitly, an arc ACB.

 

            ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ

              Return to  Na Young's Home Page

                               EMAT 6680 Home Page