Two tangents are drawn to a circle from an external point. Consider the three statements: (1) The tangents subtend equal angles at the centre of the circle. (2) The tangents drawn from the external point to the circle are equal in length. (3) The tangents are equally inclined to the segment joining the centre of the circle to that external point. Which of the following options about these tangents is correct?

Introduction / Context:
This is a conceptual geometry question about tangents drawn to a circle from an external point. Properties of tangents are very important in school geometry and in aptitude exams because they lead to many angle and length relationships used in further problem solving. Here we are given three separate statements and asked which of them are true for tangents drawn from the same external point to a circle.

Given Data / Assumptions:

Two tangents are drawn to a circle from one external point P.
Statement (1) claims that these tangents subtend equal angles at the centre of the circle.
Statement (2) claims that the tangents drawn from the external point to the circle are equal in length.
Statement (3) claims that these tangents are equally inclined to the line segment joining the centre to the external point P.
We assume the usual Euclidean geometry rules for circles and tangents.

Concept / Approach:
A key property of tangents from an external point is that the two tangents to a circle are equal in length. In addition, the radii drawn to the points of tangency are perpendicular to the tangents. By using symmetry and triangle congruence, we can show that the angles made at the centre by the radii to the tangency points are equal, and the angles between each tangent and the line joining the external point to the centre are also equal. So we examine each statement using these standard results.

Step-by-Step Solution:
Let the circle have centre O and the external point be P. Tangents touch the circle at points A and B, so PA and PB are tangents. By a basic theorem, tangents drawn from an external point to a circle are equal, so PA = PB, which supports statement (2). Draw radii OA and OB; these are perpendicular to the tangents at A and B respectively, so OA is perpendicular to PA and OB is perpendicular to PB. Consider triangles OAP and OBP. OP is common, OA = OB (both radii), and PA = PB (tangents from P). By side side side congruence, triangle OAP is congruent to triangle OBP. Therefore corresponding angles at the centre, angle AOB and angle BOA, are equal, confirming statement (1). Also, the angles between OP and each tangent, angle AOP and angle BOP, are equal, showing that the tangents are equally inclined to OP, which validates statement (3).

Verification / Alternative check:
The configuration is symmetric about the line OP. If we conceptually reflect triangle OAP across OP we get triangle OBP, meaning every geometric feature on one side has an identical counterpart on the other side. This symmetry guarantees equal angles at the centre and equal inclination of tangents to OP, as well as equal tangent lengths. Hence all three statements are consistent with standard circle theorems.

Why Other Options Are Wrong:
Option 1 and 2 only omits statement (3), but we have seen that the congruent triangles imply equal inclination of each tangent to OP, so statement (3) is also true.
Option 2 and 3 only omits the equal angle at the centre, yet congruence of triangles clearly forces the central angles formed by OA and OB to be equal.
Option 1 and 3 only omits the equal tangent lengths, which is one of the most fundamental theorems regarding tangents from an external point, so that omission makes the option incomplete.

Common Pitfalls:
Some learners incorrectly think that only the tangent lengths are equal and do not realise the stronger symmetry that also forces equal central angles.
Others confuse the phrase equally inclined with equal length and treat statements (2) and (3) as duplicates, which they are not.
Another error is to misinterpret the position of the external point and not draw the radii to the points of tangency, which makes it harder to visualise the congruent triangles and leads to wrong conclusions.

Final Answer:
All three statements are true; the correct choice is 1, 2 and 3.