1. which theorem explains why the circumcenter is equidistant from the vertices of a triangle? 2.which of the statements about the picture below are true? choose all that apply 3.if the circumcenter of a triangle lies on the triangle, then what type of triangle is it? 4. find the coordinates of the circumcenter of abc with the vertice a(0,0), b(3,0), and c(3,2). anyone have these answers to this test? ? !

1-The circumcenter refers to the central point or focal point of the circle which experiences the three vertices of the triangle. Review that all radii of a circle are congruent, i.e. equivalent to each other. So this is the reason the circumcenter is equidistant from the vertices of the triangle. The perpendicular bisectors are used to form the circumcenter, so the concurrency of perpendicular bisector theorem  also explains.
2- The picture is not given.
3-The answer is right triangle. 
In a right triangle, midpoint of hypotenuse is at equal distance from all the 3 vertices. So that is focal point of the circle going through all its 3 vertices. A right-angled triangle is a triangle which have one right angle. The connection between the sides and points of a right triangle is the reason for trigonometry. The side which lies as the opposite to the right angle is known as the hypotenuse. 
4- The coordinates of the circumcenter of ABC with the vertices A(0,0), B(3,0), and C(3,2) is (1.5,1) 
For the given triangle, vertex A lies on starting point; Vertex B lies on x-axis; and vertex C lies on hold parallel to y-axis. ==> AB along x-axis and BC opposite to AB. So the triangle ABC is a right triangle with its vertex B = 90 deg and AC has the hypotenuse. For a right triangle its circumcentre is the midpoint of hypotenuse. Consequently here the midpoint of AC = (1.5, 1), is the circumcenter of the triangle ABC.