To draw the tangent at a point on a circle without knowing the centre, one method uses any chord through the point and constructs an angle equal to the inscribed angle on the opposite arc. The reason this method works is that:

1. The tangent-chord (alternate segment) theorem says the angle between a tangent and a chord at the point of contact equals the inscribed angle on the alternate arc. 2. So by drawing an inscribed angle on the opposite arc one can mark off the same angle at the chord's end. 3. The new line through the point of contact at that angle is the tangent. 4. The construction does not depend on knowing the centre. 5. Statements about a fixed 60° angle, parallel tangents or chord-as-tangent are not generally true. _Source: SCERT Kerala Std X Mathematics Part-2, Chapter 7 "Tangents" (pp 159-172, 2019 ed.)._