5 10 pts Prove If two similar triangles have the same circum

5. (10 pts) Prove: If two similar triangles have the same circumcircle, then they must be congruent. Note: recall from class that the circumcircle of a triangle is the circle that passes through the vertices of the triangle.]

Solution

If two similar triangles have the same circumcircle, then they must be congruent.

It is quite clear because if they have same circumcircle, then they have same circumcenter.

If they have same circumcircle and same circumcenter, then they have same circumradius.

Hence they have same vertices because the circumradius is the distance of vertices from circumcenter.

So if they have same vertices, then they are obviously congruent.