Problem Statement:

In an acute-angled triangle ABC, AP is the altitude. A circle is drawn with AP as its diameter, which cuts the sides AB and AC at points D and E respectively. We need to find the length of DE.

Solution:

Given: Triangle ABC with AP as the altitude, and a circle drawn with AP as its diameter.

To find: Length of DE.

Step 1: Draw the diagram

Draw a diagram of triangle ABC with AP as the altitude. Then draw a circle with AP as its diameter. Label the points where the circle intersects sides AB and AC as D and E respectively.

Step 2: Identify the key aspects

- Triangle ABC is an acute-angled triangle.
- AP is the altitude of the triangle.
- Circle with AP as its diameter intersects sides AB and AC at points D and E respectively.

Step 3: Analyze the triangle and circle

- Since AP is the altitude, it is perpendicular to BC.
- The circle with AP as its diameter passes through A, D, and E.
- AD is a radius of the circle, as it is perpendicular to the chord DE.
- AE is also a radius of the circle, as it is perpendicular to the chord DE.

Step 4: Understand the properties of the circle

- The perpendicular from the center of a circle to a chord bisects the chord.
- Therefore, AD and AE divide DE into two equal parts.

Step 5: Determine the length of DE

- Since AD and AE bisect DE, the length of DE is equal to 2 times the length of AD or AE.
- To find the length of AD or AE, we need to determine the radius of the circle.

Step 6: Use the properties of the acute-angled triangle

- In an acute-angled triangle, the altitude is shorter than the two legs.
- Therefore, AP is shorter than AB and AC.

Step 7: Apply the properties to find the length of DE

- Since AP is shorter than AB and AC, the radius of the circle (which is half of AP) is also shorter than AB and AC.
- Therefore, DE is shorter than AB and AC.
- In other words, DE is the shortest distance between points D and E on sides AB and AC respectively.

Step 8: Conclusion

- Since DE is the shortest distance between points D and E on sides AB and AC respectively, we can conclude that DE is the perpendicular bisector of AP.
- Therefore, DE is equal to 2 times the radius of the circle, which is equal to 2 times the length of AD or AE.

Hence, the length of DE is equal to 2 times the length of AD or AE.