Circle proof question?

Question

http://www.mathwarehouse.com/geometry/circle/images/two-secants/diagram-intersection-of-secants.gif

the circle and triangle look like that, only A and C are connected, and X and Z are connected. PS: dont pay attention to any of the letters or measurements other than A, C, X and Z. The X in the angle is also not there, but label the top of the triangle B.

Given: Secants BA, BCand XA is congruent to ZC.

Prove: triangle ABC is an isosceles traingle.

Please help!!!!!!!!!!!!!!!!

Answer 1

Boy, this one is really garbled. Forget everything except that the lengths of AX and CZ are the same. That's all you need. Call the center of the circle point O, and the midpoints of AX and CZ, F and G respectively. Draw lines OF, OX, OG, OZ. Because OX = OZ, and FX = GZ, the right triangles formed are the same, and therefore OF = OG. Draw line OB, thus forming 2 right triangles where OF = OG, and OB = OB. Therefore FB = GB, from which AB = CB, which proves that it is an isoceles triangle.