If two equal-chords of a circle intersect within the circle, then the line joining the point of intersection to the center will not make equal angles with the chords.

False

Concept:

1. Equal Chords: If two chords in a circle are equal in length, they are also equidistant from the center of the circle. This means that the perpendicular distance from the center of the circle to each of the chords will be the same.
2. Properties of Equal Chords: If two chords of a circle intersect within the circle, the intersection point will have symmetrical properties with respect to the center of the circle. This is because the chords are equal and equidistant from the center, making the setup symmetric.
3. Angle Property: If two chords of a circle intersect within the circle, and we draw the center of the circle (denoted as O) and the point of intersection of the chords (denoted as T), we are interested in the angles between the line OT (from the center to the intersection point) and the two chords.

Proof:

1. Given: Two equal chords intersect at point T inside the circle. The center of the circle is O.
2. Since the chords are equal, they must be at the same distance from the center of the circle, meaning the perpendiculars dropped from the center (O) to both chords will be the same in length. Therefore, we have:
   • OTV = OTU, where T is the point of intersection of the chords, and V and U are the points where the chords meet the circle.
3. Triangles Formed: If you join the center O to the points of intersection V and U, you form two triangles, △OVT and △OUT.

4. By the Hypotenuse-Leg (HL) Theorem:

   • OV = OU (because the chords are equal).
   • OT = OT (common side).
   • ∠OVT = ∠OUT (since both chords are at equal angles from the center).

   By the HL postulate, the two triangles △OVT and △OUT are congruent.

5. By CPCTC (Corresponding Parts of Congruent Triangles are Congruent), we have:
   • ∠OTV = ∠OTU.