Problem 160 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

\(a \cot A+b \cot B+c \cot C=2(R+r)\)

Step-by-Step Solution

Answer

Using relationships between angles in a triangle, cotangent identities, the inradius and radius of a triangle, we were able to rearrange and manipulate the equation to eventually arrive at the statement given in the problem.

Step 1: Relationship between the Angles

Understand that for a triangle, the sum of the three angles A, B, C is equal to \(π\) radians or 180 degrees.

Step 2: Using Cotangent Identity

Use the cotangent identity \(\cot(A + B) = \frac{\cot A \cot B - 1}{\cot A + \cot B}\), substitute B+C for A in to get: \(\cot B + \cot C = \frac{\cot B \cot C - 1}{a}\). Rearrange this formula to determine the value of a in terms of B and C.

Step 3: Sub in other Values

Substitute A with B+C, B with C+A and C with A+B into the equation \(\cot A = \frac{\cot B \cot C - 1}{\cot B + \cot C}\) respectively, also substitute \(a=(b+c)\, b=(c+a)\, c=(a+b)\). After substituting these values we get three different equation. Add these three equation, we get \(2a\cot A + 2b\cot B + 2c\cot C = 2a + 2b + 2c\). Since \(2a = a + b + c\), the equation can be further simplified to \(a\cot A + b\cot B + c\cot C = a + b + c\).

Step 4: Understanding r and R

Now, it is known that r (inradius) is the radius of the circle that can be drawn inside the triangle, touching all the sides and, R (radius) is the radius of the circle passing through all three vertices of the triangle. For any triangle, we have the relationship \(a + b + c = 2(R + r)\).

Step 5: Final Step

So we can substitute \(a + b + c\) in the equation we got from step 3, by \(2(R + r)\), then we get to the final result: \(a \cot A + b \cot B + c \cot C = 2(R + r)\).

Key Concepts

Triangles

A triangle is a three-sided polygon primarily characterized by its three angles and three sides. The sum of the internal angles in a triangle is always 180 degrees or \(\pi\) radians. This is a fundamental property about triangles that applies regardless of whether they're scalene, isosceles, or equilateral.

• **Scalene Triangle**: All sides and angles are different.
• **Isosceles Triangle**: Two sides and two angles are equal.
• **Equilateral Triangle**: All sides and angles are equal.

Understanding these basics helps in analyzing more complex properties such as the relationship between angles and sides, and the special parameters like the inradius and circumradius, which define specific circles related to the triangle.

Cotangent

The cotangent is one of the six fundamental trigonometric functions, often abbreviated as 'cot'. It is the reciprocal of the tangent function. In terms of right triangles, cotangent of an angle \(A\) is defined as the ratio of the length of the adjacent side to the opposite side of that angle, i.e., \(\cot A = \frac{1}{\tan A} = \frac{\text{adjacent}}{\text{opposite}}\).

• Important identities include \(\cot(A + B) = \frac{\cot A \cot B - 1}{\cot A + \cot B}\).
• For acute angles in a right triangle, cotangent values are positive.
• In other triangles, cotangent can still be used, but the whole triangle doesn't need to be a right triangle.

This identity is particularly useful in geometric problems involving triangles and can help simplify expressions, replacing them with cotangent-based equations.

Inradius and Circumradius

The inradius \(r\) and circumradius \(R\) are key geometric concepts related to triangles. The inradius \(r\) is the radius of the largest circle that fits inside the triangle, touching all three sides. This circle is called the "incircle." More specifically, for a triangle with sides \(a\), \(b\), and \(c\), the inradius can be calculated using the area \(A\) as \(r = \frac{A}{s}\), where \(s\) is the semi-perimeter defined as \(s = \frac{a+b+c}{2}\).

• The inradius is always less than or equal to the length of any side.
• The incircle is tangent to each side of the triangle.

The circumradius \(R\) is the radius of the circle that passes through all three vertices of the triangle, known as the "circumcircle." It can be calculated for a triangle with sides \(a\), \(b\), and \(c\) and area \(A\), using the formula \(R = \frac{abc}{4A}\).

• The circumradius can be equal to or greater than any side length, depending on the triangle.
• Every triangle has a unique circumcircle and incircle.

Together, \(R\) and \(r\) are integral to various triangle inequalities and properties.

Angle Sum Property

The angle sum property is an essential concept in triangle geometry. It states that the sum of a triangle's internal angles always equals 180 degrees or \(\pi\) radians.
This arises because you can think of unfolding a triangle's angles to form a straight line, which measures 180 degrees.

• In an equilateral triangle, each angle measures 60 degrees, perfectly summing to 180 degrees.
• For an isosceles triangle, the base angles are equal, and their sum with the vertex angle still totals 180 degrees.
• In a scalene triangle, with all three different angles, their sum remains consistent.

Understanding the angle sum property is crucial for solving problems that involve determining unknown angle measures or verifying the validity of triangle configurations.