If in ABC, H is the orthocentre, AD, BE and CF are the altitudes, then what is HD in terms of Circumradius R, and angle B and angle C 

To determine the distance \( HD \) in terms of the circumradius \( R \) and the angles \( B \) and \( C \) in triangle \( ABC \) where \( H \) is the orthocenter, let’s analyze the problem:

### Problem Details Recap:
- Triangle \( ABC \) with orthocenter \( H \).
- \( AD \), \( BE \), and \( CF \) are the altitudes from vertices \( A \), \( B \), and \( C \) respectively, meeting the opposite sides at points \( D \), \( E \), and \( F \).
  
We need to express \( HD \) in terms of \( R \), \( \angle B \), and \( \angle C \).

### Key Concepts and Formula:

In a triangle \( ABC \):
1. The circumradius \( R \) is given by:
   \[
   R = \frac{a}{2 \sin A} = \frac{b}{2 \sin B} = \frac{c}{2 \sin C}
   \]
   where \( a = BC \), \( b = AC \), and \( c = AB \).

2. The orthocenter \( H \) is the intersection of the altitudes.

3. The distance \( HD \) (from \( H \) to the foot of the altitude \( AD \)) is given by the following formula:
   \[
   HD = 2R \cos B \cos C
   \]

### Explanation:

The formula \( HD = 2R \cos B \cos C \) is derived using the geometry of the triangle, leveraging the properties of the orthocenter and circumradius.

- The altitude \( AD \) is perpendicular to \( BC \).
- The projection of \( H \) on \( BC \) gives \( HD \), which can be calculated using the circumradius \( R \) and the angles at \( B \) and \( C \).

### Final Answer:
\[
HD = 2R \cos B \cos C
\]

This is the required expression for \( HD \) in terms of \( R \), \( \angle B \), and \( \angle C \).

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Do you need more details or have any further questions?

Here are some related questions you might find interesting:

1. How does the distance \( HE \) relate to \( R \), \( \angle A \), and \( \angle C \)?
2. What is the length of the altitude \( AD \) in terms of \( R \) and \( \angle A \)?
3. How does the circumradius relate to the inradius of the triangle?
4. What is the position of the orthocenter when the triangle is obtuse?
5. How is the nine-point circle related to the orthocenter?

**Tip:** Understanding the relationship between triangle centers like the circumcenter, orthocenter, and centroid is key to solving advanced geometry problems efficiently.

Learn how to calculate the distance HD from orthocenter H to altitude AD in triangle ABC using circumradius R and angles B and C. Detailed step-by-step solution and formula derivation provided.

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