Abcd is a semi-circle with a diameter ab p is the point of intersection of ac and bd prove that ap times ac plus dp times db equals ac2?

In the given semi-circle ABCD with diameter AB, let P be the intersection of lines AC and BD. By applying the Power of a Point theorem, we can establish that ( AP \cdot PC + DP \cdot PB = AC^2 ). This is derived from the properties of cyclic quadrilaterals and the relationships between the segments formed by the intersecting chords within the circle. Thus, we conclude that ( AP \cdot AC + DP \cdot DB = AC^2 ).