prove that only two tangents can be drawn from an external point of a circle

Let's draw an isosceles triangle ABC, with base BC and equal sides AB and AC. Draw an another isoscles triangle DBC with equal sides DB and DC below the triangle ABC such that angle ABD = 90°
Now, In triangle ABD & ACD,
AB = AC [equal sides of tri. ABC]
DB = DC [equal sides of tri. DBC]
AD = AD [common]
so, 🔺 ABD =~ 🔺 ACD [=~ is congruency sign]
so, angle ABD = angle ACD = 90° ....(1) [cpct]
Now, taking A as centre and AB as radius, we can draw a circle (A,AB).
This circle will pass through AC as well(AC = AB)
Now, DC and DB are 2 lines passes through C and B respectively through which circle(A,AB) also passes.
so, DB and DC touch the circle.
Also, angle ABD = 90° = Angle ACD [from (1)]
so, from all these, we conclude that DC and DB are 2 tangents line of circle(A,AB) which have drawn through an external point D.
so, 2 tangents can be drawn through an external point of circle.
Now, let 3 tangents can be drawn through that point to the circle which are DB,DC and DE.
Then, these tangents must be equal to each other these all tangents will intersect the circle (A,AB) on 3 points B,C and E respectively.
Now, as 3 tangents are equal.
so, we can say that there is a circle of centre D which passes through these 3 points B,C and E.
Also, circle(A,AB) passes through these 3 points.
so, these 2 circle passes through 3 non-colinear points which is impossible as there is an unique circle passes through 3 non-colineae points.
so, there can't exist 3 tangents of a circle passes through the same external point.(here D)
so, only 2 tangents can draw through an external point of circle.

(Hope you will interpret it. but please draw a diagram according to the said above in the solution to interpret it much clearly)