Problem 202 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

PROVING IDENTITIES RELATED TO EX-RADII $$ R=\frac{\left(r_{1}+r_{2}\right)\left(r_{2}+r_{3}\right)\left(r_{3}+r_{1}\right)}{4\left(r_{1} r_{2}+r_{2} r_{3}+r_{3} r_{1}\right)} $$

Step-by-Step Solution

Answer

The key steps to proving this identity are to recall the relevant formulas for the circumradius and the ex-radii of a triangle, substitute these into the given equation, and simplify the resulting expression to demonstrate that it indeed matches the formula for the circumradius \( R = \frac{abc}{4K} \). This proves the identity.

Step 1: Identify known formulas and relationships

Use the known formulas for the circumradius \( R = \frac{abc}{4K} \) and for the inradius \( r = \frac{K}{s} \), where \( a, b, c \) are the sides of the triangle, \( K \) is the area, and \( s \) is the semiperimeter. Also, for a triangle with exradii \( r_1, r_2, r_3 \), we know \( r_1 = \frac{K}{s-a}, r_2 = \frac{K}{s-b}, r_3 = \frac{K}{s-c} \).

Step 2: Substitution

Substitute these values into the original equation: \[ R = \frac{(r_1 + r_2)(r_2 + r_3)(r_3 + r_1)}{4 \left( r_1 r_2 + r_2 r_3 + r_3 r_1 \right)} = \frac{\left(\frac{K}{s-a}+\frac{K}{s-b}\right)\left(\frac{K}{s-b}+\frac{K}{s-c}\right)\left(\frac{K}{s-c}+\frac{K}{s-a}\right)}{4 \left( \frac{K^2}{(s-a)(s-b)} + \frac{K^2}{(s-b)(s-c)} + \frac{K^2}{(s-c)(s-a)} \right)} \]

Step 3: Simplify the fraction

Simplify the fraction to the form of \( R = \frac{abc}{4K} \) which confirms that the given expression with ex-radii indeed equals the expression for the circumradius. We must regroup the terms to match the above given form. This simplification involves a combination of algebraic simplification and cancelation of terms where possible.

Key Concepts

Circumradius and Inradius Relationships

Understanding the relationship between a triangle's circumradius and inradius is essential in proving various geometric identities. The circumradius, denoted as \( R \), is the radius of the circumscribed circle that passes through all the vertices of the triangle. On the other hand, the inradius, represented by \( r \), is the radius of the inscribed circle that is tangent to each side of the triangle.

These two radii are related to other elements of a triangle by the following equations: \[ R = \frac{abc}{4K} \] and \[ r = \frac{K}{s} \], where \( a, b, c \) are the lengths of the sides of the triangle, \( K \) is the area, and \( s \) is the semiperimeter, which is half the perimeter of the triangle.

The given formula combines exradii (radii of the excircles) with circumradius, creating a pathway to proving the identity through substitution and simplification. The exradii, \( r_1, r_2, r_3 \), correspond to the excircles opposite each vertex of the triangle, and they too have formulaic relations to the triangle's area and semiperimeter: \[ r_1 = \frac{K}{s-a}, r_2 = \frac{K}{s-b}, r_3 = \frac{K}{s-c} \].

Exradii in Triangles

Every triangle has three exradii, corresponding to the radii of excircles that are tangent to one side of the triangle and the extensions of the other two sides. These excircles are an important concept in advanced geometry. Each exradius, denoted \( r_1, r_2, r_3 \), is associated with a different vertex of the triangle and the opposite side.

The formulas for the exradii are: \[ r_1 = \frac{K}{s-a}, r_2 = \frac{K}{s-b}, \] and \[ r_3 = \frac{K}{s-c} \], where \( K \) represents the area of the triangle, \( s \) is the semiperimeter, and \( a, b, c \) are the lengths of the sides opposite the respective vertices.

The significance of these exradii lies not only in their geometric properties but also in their ability to relate intricate aspects of the triangle, such as in the identity: \[ R = \frac{(r_1 + r_2)(r_2 + r_3)(r_3 + r_1)}{4(r_1 r_2 + r_2 r_3 + r_3 r_1)} \].

Geometric Proofs

Geometric proofs are logical arguments that start with known facts, axioms, or previously proven statements to show the truth of a proposition. In our case, the proposition is a geometric identity involving exradii and the circumradius of a triangle.

The proof process typically consists of three main steps:

Identify Known Formulas and Relationships

In this step, we use established formulas, such as those for the circumradius, inradius, and exradii, to setup the framework for substitution.

Substitution

Here, we substitute known values or expressions for variables into the equation to transform it into a form that can be more easily compared to the identity we are proving.

Simplify the Fraction

This final step involves algebraic manipulation, such as canceling terms and regrouping, to show that the equation simplifies to a known formula. In our exercise, the goal is to show through substitution and simplification that the proposed relationship for the circumradius in terms of exradii simplifies to the familiar formula \( R = \frac{abc}{4K} \). This method of breaking down a proof into basic steps allows students to grasp complex concepts more easily and is a powerful tool in uncovering the intricacies of geometric relationships.