Let ABC be an acute angled triangle whose orthocentre is at H. If altitude from A is produced to meet the circumcircle of triangle ABC at `D` , then prove `H D=4RcosBcosC`

Explanation

To prove that \( HD = 4R \cos B \cos C \) for triangle \( ABC \) with orthocenter \( H \) and point \( D \) where the altitude from \( A \) meets the circumcircle, we can follow these steps:

### Step 1: Understand the Geometry
Let \( ABC \) be an acute-angled triangle with vertices \( A \), \( B \), and \( C \). The orthocenter \( H \) is the intersection of the altitudes of the triangle. The point \( D \) is where the altitude from \( A \) meets the circumcircle of triangle \( ABC \).

### Step 2: Draw the Diagram
Draw triangle \( ABC \) with the circumcircle. Mark the orthocenter \( H \) and the foot of the altitude from \( A \) to \( BC \) as \( N \). Extend the altitude from \( A \) to meet the circumcircle at point \( D \).

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