Let ABC a triangle with base BC. If we raise a line perpendicular to BC touching the edge A of this triangle. this will divide it into two right triangles. With bases now BD and DC.
The book says: AC^2 = BC x DC.
What i don't understand is that if we just stretch the line BD, the above formula wouldn't be true. Or am I missing something here. Please help. Thanks.
As some people commented, this can only happen if $ABC$ was a right angled triangle.
Here is proof that the formula is true:
If you have right angled triangle $ABC$, with base $BC$, and you have a perpendicular line to $BC$ touching the right angle $A$, then you end up with two triangles that are similar to the whole triangle and to each other.
Since the triangles are similar, the ratio of their corresponding sides are equal. So this means $\frac{AC}{DC} = \frac{BC}{AC} \implies AC^2 = BC*DC$.