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 +