Consider a line drawn from point P outside of circle whose centre is O as a tangent to this circle at T. (Tangents just touch a circumference but do not enter the circle.) 

Prove ∠  OTP is a right angle. 

Draw any other line from O to line PT and let M be the point where this line meets PT. and intersect circle at S

OM > OS 

OS = OT                                                                             ...(radius of circle)

∴ OM > OT 

∴ OT is shortest distance from O to line PT

∴ OT must be perpendicular to PT                     ... (shortest dist. between point and line is perpendicular)