Proof:
To prove that AB^2 * AC^2 = AP^2 * AQ^2 * 4PQ^2, we'll use the concept of similar triangles and the Pythagorean theorem.

1. Triangles ABC and APQ are similar:
- Since AP and AQ are both trisecting side BC, we can conclude that BP = PQ = QC.
- Therefore, triangles ABC and APQ have the same angles, making them similar.

2. Proving AB/AP = AC/AQ:
- Since triangles ABC and APQ are similar, we can write the following ratios:
- AB/AP = BC/PQ
- AC/AQ = BC/PQ
- Since BP = PQ and QC = PQ, we can substitute these values into the ratios:
- AB/AP = BC/BP
- AC/AQ = BC/QC
- We can see that BP = QC, so we can simplify the ratios:
- AB/AP = BC/BP = BC/QC = AC/AQ
- Therefore, AB/AP = AC/AQ.

3. Applying the Pythagorean theorem:
- From the similarity of triangles ABC and APQ, we know that AB/AP = AC/AQ.
- Let's call this common ratio k. So, AB = k * AP and AC = k * AQ.
- Using the Pythagorean theorem, we can write the following equations:
- AB^2 = AP^2 + BP^2
- AC^2 = AQ^2 + QC^2
- Since BP = QC, we can simplify the equations:
- AB^2 = AP^2 + PQ^2
- AC^2 = AQ^2 + PQ^2
- Substituting AB = k * AP and AC = k * AQ, we get:
- (k * AP)^2 = AP^2 + PQ^2
- (k * AQ)^2 = AQ^2 + PQ^2
- Simplifying these equations, we have:
- k^2 * AP^2 = AP^2 + PQ^2
- k^2 * AQ^2 = AQ^2 + PQ^2
- Rearranging the equations, we get:
- (k^2 - 1) * AP^2 = PQ^2
- (k^2 - 1) * AQ^2 = PQ^2
- Since AB/AP = AC/AQ, we know that k^2 - 1 = 3.
- Substituting this value, we have:
- 3 * AP^2 = PQ^2
- 3 * AQ^2 = PQ^2
- Multiplying both equations by 4, we get:
- 4 * AP^2 = 4 * PQ^2 / 3
- 4 * AQ^2 = 4 * PQ^2 / 3
- Therefore, AB^2 * AC^2 = AP^2 * AQ^2 * 4 * PQ^2.

Conclusion:
From the similarity of triangles ABC and APQ, and by applying the Pythagorean theorem, we have proved that AB^2 * AC^2 = AP^2 * AQ^2 *