Join point Q to tangent points A and B.

According to Alternate Segment Theorem, the angle between a tangent and a chord is equal to the angle in the alternate segment.

Then,

\angle QBD = \angle QAE

So the right \triangle QBD and \triangle QBD are similar and the ratios of corresponding sides are equal.

\dfrac{QD}{QE}=\dfrac{QB}{QA}

(1)

Similarly,

\angle QAC = \angle QBE

So the right \triangle QAC and \triangle QBE are similar and the ratios of corresponding sides are equal.

\dfrac{QC}{QE} = \dfrac{QA}{QB}

(2)

Based on (1) and (2), we get

\dfrac{QD}{QE} = \dfrac{QE}{QC}

(3)

Therefore,

QE = \sqrt{QD\cdot QC}

Now we have proved that the perpendicular to AB is the mean proportional between the other two perpendiculars