Question

prove that the tangent at any extremes of any chord makes equal angle with the chord

Answer 1

Draw a Circle.

Take any point outside the circle. Distance of the point from circle may be any distance.

Draw a tangential line from that point to circle. There could be 2 tangential line drawn from that point to any circle.

Now connect the both tangent point in the circle which makes the Chord.

Then connect those both tangents to center of the circle.

Lets consider names as follows.

Center of the Circle = O

Tangent Point in Circle = X & Y

Points consider outside the circle = P

There are two triangles you can see ΔOXP & ΔOYP

(i) OX & OY is equal and both are radius for that circle.

(ii) OP is common for both triangles.

(iii) ∠OXP & ∠OYP must be equal and it is 90° since these lines are tangent to the circle.

Since all 3 factors of triangle is same for both these triangles the angle of chord makes equal angle with Chord as follows.

∠OXP = ∠OYP

I don't know how to draw here. Let me know if further clarification is required.

Enjoy learning.