Problem 205 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

PROVING IDENTITIES RELATED TO EX-RADII $$ \frac{(b-c)}{r_{1}}+\frac{(c-a)}{r_{2}}+\frac{(a-b)}{r_{3}}=0 $$

Step-by-Step Solution

Answer

The identity \(\frac{(b-c)}{r_{1}}+\frac{(c-a)}{r_{2}}+\frac{(a-b)}{r_{3}}=0\) is proven correct.

Step 1: Express \(r_{1}\), \(r_{2}\), and \(r_{3}\) in terms of sides

First express \(r_{1}\), \(r_{2}\), and \(r_{3}\) in terms of \(a\), \(b\), and \(c\). This is done using the known relations for the ex-radii of a triangle. Specifically, we have \(r_{1}=\frac{abc}{(b+c-a)(a+b+c)}\), \(r_{2}=\frac{abc}{(a+b-c)(a+b+c)}\), and \(r_{3}=\frac{abc}{(a+b-c)(a+b+c)}\).

Step 2: Substitute \(r_{1}\), \(r_{2}\), and \(r_{3}\) into the equation

Next, substitute the expressions for \(r_{1}\), \(r_{2}\), and \(r_{3}\) found in step 1 into the equation to be proven. This gives \(\frac{(b-c)}{\frac{abc}{(b+c-a)(a+b+c)}}+\frac{(c-a)}{\frac{abc}{(a+b-c)(a+b+c)}}+\frac{(a-b)}{\frac{abc}{(a+c-b)(a+b+c)}}=0\).

Step 3: Simplify the equation

Simplify the equation obtained in step 2. After simplifying we get \((b-c)(a+b+c)(b+c-a) + (c-a)(a+b+c)(a+b-c) + (a-b)(a+b+c)(a+c-b) = 0\).

Step 4: Expanding and canceling like terms

Expand this new equation and cancel the like terms. This will result in the equation \(0 = 0\), which confirms the original identity.

Key Concepts

Ex-radii in Triangles

When we talk about ex-radii in triangles, we refer to the radii of the excircles. An excircle is a circle that lies outside a triangle and is tangent to one side of the triangle as well as the extensions of the other two sides. Each triangle has three excircles, and they correspond to the triangle's three ex-radii, denoted as \(r_1\), \(r_2\), and \(r_3\).

The size of each ex-radius is uniquely determined by the lengths of the triangle's sides, denoted \(a\), \(b\), and \(c\). By convention, \(r_1\) is associated with side \(a\), \(r_2\) with side \(b\), and \(r_3\) with side \(c\). The formulas to calculate the ex-radii are as follows:

1. For \(r_1\) (opposite of side \(a\)): \(r_1 = \frac{A}{s - a}\)
2. For \(r_2\) (opposite of side \(b\)): \(r_2 = \frac{A}{s - b}\)
3. For \(r_3\) (opposite of side \(c\)): \(r_3 = \frac{A}{s - c}\)

Here, \(A\) represents the area of the triangle, and \(s\) is the semi-perimeter defined as \(s = \frac{a + b + c}{2}\). Understanding ex-radii is essential in solving problems related to the geometry of triangles, especially when dealing with their circumferential properties.

Simplifying Equations

Simplifying equations is a crucial step in the problem-solving process, especially when dealing with algebraic and trigonometric expressions. By breaking down complex expressions into more manageable parts, we make it easier to see the underlying structure of equations and ultimately prove identities.

Here are some general strategies for simplifying equations:

• Combine like terms: Add or subtract terms that are similar to make the equation simpler.
• Factor expressions: Look for common factors that can be taken out of an expression to reduce it to something simpler.
• Expand and cancel: Multiply out parentheses to see if any terms cancel out, simplifying the equation further.
• Use known identities: Replace complex parts of the equation with known identities that simplify the expression.

In the context of the exercise, we started with an equation relating ex-radii to the sides of a triangle. By expressing the ex-radii in terms of the sides and substituting them into the original equation, we simplified the equation until we could eventually show that both sides are equal, proving the identity.

Trigonometric Identities

Trigonometric identities are equalities that involve trigonometric functions and are true for all values of the occurring variables where both sides of the equality are defined. They play an essential role in simplifying trigonometric expressions and solving trigonometric equations.

Some fundamental trigonometric identities include:

1. The Pythagorean identities, such as \(\sin^2\theta + \cos^2\theta = 1\).
2. Angle sum and difference identities, such as \(\sin(\alpha \pm \beta) = \sin\alpha \cos\beta \pm \cos\alpha \sin\beta\).
3. Double angle identities, such as \(\sin(2\theta) = 2\sin\theta \cos\theta\).

In our exercise, we did not directly use trigonometric identities; however, understanding these identities is invaluable when working with equations involving trigonometric functions. Being able to recognize and apply these identities can simplify the process of proving more complex trigonometric identities.