be equidistant from the center of the circle.
APEX!
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Be equally far from the center and create congruent angles with the tangent line at the point where the chord intersects the circle.
They must be congruent.
congruent
The triangles must be congruent.
Yes, any diameter which is perpendicular to a chord bisects said chord. This can be proved most easily with a picture, but is proved using a congruent triangle proof. Both triangles include the points at the center of the circle and the intersection of the diameter and chord. The other points should be the endpoints of the chord. They are congruent by hypotenuse leg; it was given that they are right triangle by the "perpendicular", the "leg" is the segment between the center of the circle and the intersection, and it is equal in both triangles because it is the same segment in both triangles. The hypotenuses are equal because both are radii of the circle. Because the triangles are congruent, their sides must be so the two halves of the chord are congruent, and therefore the chord is bisected by the diameter.
Yes. A circle is defined as the set of all points in a plane equidistant from a given point (the center of the circle) - hence - all points of a circle must be co-planar by definition.