# 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.

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.

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.

Now we will investigate the change of an angle C.

Move the point C on the circle O.

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.

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