In AABC, D and E are the mid-points of the sides AB and AC respectively, CD and BE intersect at P with BPC = 90°. Suppose BD = 8√5 and CE =6√5

In the triangle \( \triangle ABC \), with points \( D \) and \( E \) being the midpoints of sides \( AB \) and \( AC \) respectively, and given that lines \( CD \) and \( BE \) intersect at point \( P \) such that \( \angle BPC = 90^\circ \), along with the lengths of \( BD = 8\sqrt{5} \) and \( CE = 6\sqrt{5} \), we can analyze and deduce certain properties of this configuration.

### Step 1: Midpoint Theorem
Since \( D \) and \( E \) are midpoints of sides \( AB \) and \( AC \), by the midpoint theorem:
- \( BD = \frac{1}{2} AB \)
- \( CE = \frac{1}{2} AC \)

From the given:
- \( BD = 8\sqrt{5} \), so \( AB = 2 \times BD = 16\sqrt{5} \)
- \( CE = 6\sqrt{5} \), so \( AC = 2 \times CE = 12\sqrt{5} \)

Thus, we now know the lengths of \( AB \) and \( AC \).

### Step 2: Coordinate Geometry Approach (Optional)
One potential approach to solve this fully is using coordinate geometry:
1. Place \( B \) at \( (0, 0) \) and \( C \) at \( (x_1, y_1) \), with point \( A \) at \( (x_A, y_A) \). Then calculate the midpoints \( D \) and \( E \).
2. Find the equations of the lines \( CD \) and \( BE \) and solve for the intersection point \( P \), using the condition \( \angle BPC = 90^\circ \).

This involves more algebra and calculus, so let me know if you want to proceed with this approach.

### Step 3: Conclusion & Understanding
This configuration involves significant geometric relationships, including the fact that the lines \( BE \) and \( CD \) are medians of the triangle and intersect at the centroid, which divides the medians in a 2:1 ratio. The additional condition of \( \angle BPC = 90^\circ \) imposes further restrictions, likely implying perpendicularity due to orthogonality.

Let me know if you want further details, specific calculations, or any geometric insights into this problem!

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Here are five related questions to deepen understanding:
1. How does the midpoint theorem help simplify problems involving medians in triangles?
2. How can you prove that medians of a triangle intersect at a single point (centroid)?
3. What role does the 90-degree angle condition play in triangle geometry problems?
4. Can we solve the problem using vector geometry instead of coordinate geometry?
5. How does the centroid divide the medians of a triangle, and why is this property important?

**Tip**: In any triangle, the centroid is the balance point, dividing each median in a ratio of 2:1 from the vertex.

Solve a complex triangle geometry problem where medians intersect at 90°, involving key concepts like the midpoint theorem, properties of medians, and centroid division. Learn how BD = 8√5 and CE = 6√5 impact the geometry of the triangle. Suitable for students in Grades 10-12.

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