Consider the following statements in respect of an equilateral `DeltaABC.`
I. There is a point P inside the `DeltaABC`, such that each of its sides subtends an angle of `120^@` at P. II. There is a point P inside `DeltaABC`, such that the `DeltaPBC`, iis obtuse angled and A is the orthocentre of `DeltaPBC`. Which of the above statements is/are correct ?

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

To solve the problem, we need to analyze the two statements regarding the equilateral triangle \( \Delta ABC \).

### Step 1: Analyze Statement I
The first statement claims that there exists a point \( P \) inside the triangle \( \Delta ABC \) such that each side of the triangle subtends an angle of \( 120^\circ \) at point \( P \).

- **Understanding the Geometry**: In an equilateral triangle, all angles are equal to \( 60^\circ \). When we consider point \( P \) inside the triangle, the angles subtended by the sides at point \( P \) can be calculated using the property of angles in a triangle.
- **Finding Angles**: If point \( P \) is positioned such that it subtends \( 120^\circ \) at each side, we can visualize this. The angles around point \( P \) must sum to \( 360^\circ \). If each side subtends \( 120^\circ \), then the total subtended angle would be \( 3 \times 120^\circ = 360^\circ \), which is valid.
- **Conclusion for Statement I**: Therefore, Statement I is correct.

### Step 2: Analyze Statement II
The second statement claims that there exists a point \( P \) inside \( \Delta ABC \) such that triangle \( \Delta PBC \) is obtuse-angled and point \( A \) is the orthocenter of triangle \( \Delta PBC \).

- **Understanding Orthocenter**: The orthocenter of a triangle is the point where the altitudes intersect. In an equilateral triangle, the orthocenter coincides with the centroid and the circumcenter.
- **Conditions for Obtuseness**: For triangle \( \Delta PBC \) to be obtuse, one of its angles must be greater than \( 90^\circ \). If \( A \) is the orthocenter of triangle \( \Delta PBC \), then the altitude from \( A \) to side \( BC \) must intersect at a point within the triangle.
- **Geometric Impossibility**: However, since \( A \) is a vertex of the equilateral triangle, it cannot serve as the orthocenter of triangle \( \Delta PBC \) while maintaining the obtuse condition. The orthocenter of triangle \( PBC \) cannot be at vertex \( A \) because it would contradict the properties of the triangle.
- **Conclusion for Statement II**: Therefore, Statement II is incorrect.

### Final Conclusion
Only Statement I is correct, while Statement II is incorrect.

### Summary of the Solution
- Statement I is correct: There exists a point \( P \) inside \( \Delta ABC \) such that each side subtends an angle of \( 120^\circ \) at \( P \).
- Statement II is incorrect: There cannot be a point \( P \) such that \( A \) is the orthocenter of triangle \( PBC \) while \( PBC \) is obtuse.

### Final Answer
**Only Statement I is correct.**

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