In this video transcription, the goal is to prove that BA dot BC equals one quarter of (BD squared minus AC squared), where BCD is a parallelogram with a center at point H. The speaker initially tries to approach the problem from both sides but finds that it doesn't work well. Instead, they recommend starting from the left side and trying to simplify as much as possible, then doing the same for the right side, and finally comparing the two to see if they can be shown to be equal. The speaker also notes that the fact that H is the center of the parallelogram is important and should be used. By treating BD squared minus AC squared as a scalar function of AC, the speaker is able to simplify the expression to 4 times BO squared minus AO squared, which can then be compared to BO squared minus AO squared plus OC squared minus OA squared. Finally, it is shown that if BA dot BC equals 0, then BD equals AC, proving that ABCD is a rectangle. The main takeaway from this video is the method of approaching a problem from both sides and simplifying as much as possible before comparing the two expressions.