Let's analyze the given problem step-by-step:

Given:
- Triangle \(ABC\) with point \(P\) on side \(AC\)
- \(3 |AP| = |PC|\)
- \(\angle BPC = 60^\circ\)
- \(\angle ACB = 75^\circ\)

We are asked to find \(\angle BAC\).

First, let's assign variables to the lengths:
- Let \( |AP| = x \)
- Then \( |PC| = 3x \)

This makes \( |AC| = |AP| + |PC| = x + 3x = 4x \).

Since \(\angle BPC = 60^\circ\), we can use the properties of triangle geometry and the given angles to determine \(\angle BAC\).

Step-by-step approach:

1. **Use the given angles**:
   - \(\angle ACB = 75^\circ\)
   - Let \(\angle BAP = \alpha\) and \(\angle PAC = \beta\)

2. **Use the triangle angle sum property**:
   - In \(\triangle APC\), we have \(\angle APC + \angle BAP + \angle PAC = 180^\circ\)
   - We know \(\angle APC = 60^\circ\), so:
     \[
     60^\circ + \alpha + \beta = 180^\circ \implies \alpha +

Explore how to solve a triangle geometry problem involving point P on side AC of triangle ABC. Understand angle relationships and apply the triangle angle sum theorem to find angle BAC. Suitable for high school students.

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