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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