#6 PQ perpendicular to the tangent line at the point of tangency
#6 PQ perpendicular to the tangent line at the point of tangency
Given line l tangent to circle centered at Q at point P. Prove PQ is perpendicular to l.
Assume that PQ is not perpendicular to l. Let R be a point on l such that QR is perpendicular to l. The QR<QP since the perpendicular is the shortest distance between Q and l. Since l is tangent at P then that implies that R is outside the circle. Then QR intersects the circle at some point X and QX<QR. Notice that QX=QP then QP<QR this contradicts the assuption that QR<QP. Therefore QP is perpendicular to l.