Question

In \(\triangle ABC\), \(bc - r_2 r_3 = \)

• A. \( rr_1 \)
• B. \( rr_2 \)
• C. \( r_1 \)
• D. \( a r_1 \)

Hint:

In triangle geometry, when dealing with exradii and the inradius, understanding the relationships between the side lengths, area, and the various radii is key to simplifying expressions and solving complex geometric identities.

Answer 1

The Correct Option is A

Step 1: Recognize that \( bc - r_2 r_3 \) involves geometrical properties related to the sides and the radii of the excircles of the triangle. Here, \( b \) and \( c \) are the lengths of two sides of the triangle, and \( r_2 \) and \( r_3 \) represent the exradii opposite vertices \( B \) and \( C \), respectively.
Step 2: The expression suggests a relationship between the product of the side lengths and the product of the exradii. This is a well-known geometric identity that arises from the triangle's geometry. Specifically, this identity connects the semiperimeter, the exradii, and the inradius. 
Step 3: The equation \( bc - r_2 r_3 = r r_1 \) holds true because of the geometric properties of the triangle, where \( r_1 \) is the inradius, and the exradii \( r_2 \) and \( r_3 \) are related to the area and the semi-perimeter of the triangle. 
Step 4: This identity is a special case of a more general relationship in triangle geometry involving the sides and radii. The result connects the inradius and the product of the side lengths and excircles in a way that simplifies to \( rr_1 \).