Which of the following statements is/are correct?
The intersections of the angle bisectors of the quadrilateral are the vertices of the cyclic quadrilateral
If diagonals of a cyclic quadrilateral perpendicularly intersect at point P, then the line through P, perpendicular to any side bisects the opposite side.
The product of the lengths of the two diagonals of a cyclic quadrilateral is equal to the sum of the products of opposite sides.

Options

A. (i) and (ii)
B. (ii) and (iii)
C. (i) and (iii)
D. all of these

Answer

To determine which of the statements regarding cyclic quadrilaterals are correct, we will analyze each statement step by step.

### Step 1: Analyze the First Statement
**Statement:** The intersections of the angle bisectors of the quadrilateral are the vertices of the cyclic quadrilateral.

**Analysis:**
- A cyclic quadrilateral is defined as a quadrilateral whose vertices lie on the circumference of a circle.
- The angle bisectors of a quadrilateral do not necessarily intersect at points that can be the vertices of a cyclic quadrilateral.
- The intersection of the angle bisectors of a quadrilateral does not guarantee that these points will form a cyclic quadrilateral.

**Conclusion:** This statement is **incorrect**.

### Step 2: Analyze the Second Statement
**Statement:** If diagonals of a cyclic quadrilateral perpendicularly intersect at point P, then the line through P, perpendicular to any side bisects the opposite side.

**Analysis:**
- In a cyclic quadrilateral, if the diagonals intersect at point P and are perpendicular, then any line drawn through P that is perpendicular to one side of the quadrilateral will indeed bisect the opposite side.
- This is based on the properties of cyclic quadrilaterals and perpendicular bisectors.

**Conclusion:** This statement is **correct**.

### Step 3: Analyze the Third Statement
**Statement:** The product of the lengths of the two diagonals of a cyclic quadrilateral is equal to the sum of the products of opposite sides.

**Analysis:**
- This is a well-known theorem in geometry concerning cyclic quadrilaterals.
- Mathematically, if we denote the lengths of the diagonals as AC and BD, and the sides as AB, BC, CD, and DA, then the relationship can be expressed as:
\[ AC \cdot BD = AB \cdot CD + AD \cdot BC \]
- This relationship holds true for all cyclic quadrilaterals.

**Conclusion:** This statement is **correct**.

### Final Conclusion
Based on the analysis:
- The first statement is incorrect.
- The second statement is correct.
- The third statement is correct.

Thus, the correct statements are the **second and third statements**.

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