The diagonals AC and BD of a cyclic quadrilateral ABCD intersect each other at the point P. Then,it is always true that

Options

A. AP. CP = BP. DP
B. AP.BP = CP. DP
C. AP. CD = AB. CP
D. BP. AB = CD.CP

Answer

To solve the problem regarding the cyclic quadrilateral ABCD and the intersection of its diagonals AC and BD at point P, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding Cyclic Quadrilaterals**:
A cyclic quadrilateral is a four-sided figure where all vertices lie on the circumference of a circle. This implies that opposite angles of the quadrilateral are supplementary.

2. **Identifying the Angles**:
When diagonals AC and BD intersect at point P, we can denote the angles formed at P. Let:
- ∠APB = x
- ∠CPD = y
- ∠BAP = z
- ∠DCP = w

3. **Using the Property of Angles in Cyclic Quadrilaterals**:
In a cyclic quadrilateral, the angles subtended by the same arc are equal. Thus, we have:
- ∠APB = ∠CPD (angles subtended by arc AB)
- ∠BAP = ∠DCP (angles subtended by arc CD)

4. **Establishing Angle Relationships**:
From the above properties, we can say:
- x = y (since they subtend the same arc)
- z = w (since they subtend the same arc)

5. **Conclusion**:
Therefore, we can conclude that:
- The triangles APB and DPC are similar by the AA (Angle-Angle) criterion, since they have two pairs of equal angles.

### Final Statement:
Thus, it is always true that the angles formed at the intersection of the diagonals of a cyclic quadrilateral maintain the property of equality based on the arcs they subtend.