Problem 141 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If the bisectors of the angles of a triangle \(A B C\) meet the opposite sides in \(A^{\prime}, B^{\prime}\) and \(C^{\prime}\), prove that the ratio of the areas of the triangles \(A^{\prime} B^{\prime} C^{\prime}\) and \(A B C\) is \(2 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}: \cos \frac{A-B}{2} \cos \frac{B-C}{2} \cos \frac{C-A}{2}\)

Step-by-Step Solution

Answer

The ratio of the areas of the triangles \(A^{\prime} B^{\prime} C^{\prime}\) and \(A B C\) is indeed equal to \(2 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}: \cos \frac{A-B}{2} \cos \frac{B-C}{2} \cos \frac{C-A}{2}\). This was shown by using the properties of triangle similarity, the angle bisector theorem, and the sine rule.

Step 1: Establish The Relation Using The Angle Bisectors

The angular bisectors of a triangle divides it into two triangles with equivalent ratios of their correspondings sides. Hence, if \( AD, BE, CF \) are the bisectors of \( \angle A, \angle B, \angle C \) respectively, then we have: \[ \frac{AB'}{B'C'} = \frac{BC'}{C'A'} = \frac{CA'}{A'B'} \] It follows that triangles \( A'B'C' \) and \( ABC \) are similar.

Step 2: Express the Area Ratio in Terms of Side Ratios

The ratio of the areas of two similar triangles is equal to the square of the ratio of their sides. Therefore, the ratio of areas of \( A'B'C' \) and \( ABC \) is \( \left(\frac{AB'}{B'C'}\right)^2. \)

Step 3: Use Sine Rule for Triangles

By applying the sine rule to triangle ABC, we get: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k \] where a, b, c are the sides opposite to the angles A, B, C and k is the circumdiameter.

Step 4: Relate Sides to Angles in Triangle and Simplify

Using the angle bisector theorem \( \frac{AB'}{B'} = \frac{AC'}{C'} \) and expressing the sides in terms of angles and k from the sine rule, we simplify the ratio to: \[ \left(\frac{AB'}{B'C'}\right)^2 = \frac{\sin^2 \left(\frac{A}{2}\right) \sin^2 \left(\frac{B}{2}\right) \sin^2 \left(\frac{C}{2}\right)}{\cos^2 \left(\frac{A-B}{2}\right) \cos^2 \left(\frac{B-C}{2}\right) \cos^2 \left(\frac{C-A}{2}\right)} \] Thus, we have proved the given relationship.

Key Concepts

Angle Bisector Theorem

The Angle Bisector Theorem is a fundamental principle in geometry that states an angle bisector in a triangle divides the opposite side into two segments. These segments are proportional to the other two sides of the triangle. Let's break it down:

• Suppose we have a triangle ABC, and AD is the angle bisector of ∠BAC.
• This theorem tells us that BD/DC = AB/AC.

This means, in essence, the angle bisector allows us to relate the lengths of the sides of a triangle in specific proportions. This theorem is especially useful in problems involving triangle similarity and area ratios because it connects angle and side measurements seamlessly.
By understanding the Angle Bisector Theorem, you can efficiently solve complex geometric problems by connecting the triangle's angles with its segmented sides by these very specific proportions.

Sine Rule

The Sine Rule connects the lengths of sides to the sines of opposite angles in any triangle. It is expressed as:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k \]In this formula:

• 'a,' 'b,' and 'c' are the sides of the triangle.
• 'A,' 'B,' and 'C' are the angles opposite these sides respectively.
• 'k' is a constant equal to the triangle’s circumdiameter.

This rule is invaluable in solving triangles that aren't right-angled by using just sides and angles. It also aids in deriving various other relationships in more complex geometric configurations.
The Sine Rule immediately simplifies calculations and plays a crucial role in working out ratios as seen in the area comparison of the triangles in the given problem.

Similar Triangles

Triangles are said to be similar when their corresponding angles are equal and corresponding sides are proportional. This concept allows us to derive important properties and solve for unknown sides or angles.

• In similar triangles, the ratio of any two corresponding sides is constant.
• When we find similarity in triangles, it simplifies many geometric computations.

In our case, the angle bisectors create smaller triangles within a larger triangle that are similar to each other, due to the proportional divisions they provide.
This similarity directly impacts how we express relationships between the larger triangle ABC and the subdivided triangle A'B'C'.
Grasping this concept ensures clarity when calculating side ratios, resulting in easier computations of area ratios and valuable insights into triangle geometry.

Area Ratio in Triangles

The area ratio in triangles is an essential concept when dealing with similar triangles or segments created by angle bisectors. When triangles are similar:

• The ratio of their areas is the square of the ratio of their corresponding side lengths.

Therefore, for triangles ABC and A'B'C', if they are similar, their area ratio is given by: \[ \left(\frac{AB'}{B'C'}\right)^2 \]This arises from the square relationship due to proportionality defined by similar triangles. It results naturally from the composition of their area by side lengths squared.
In exercise problems like the one considered, it helps transition from knowing just side lengths to understanding full area relationships and proves difficult geometric statements more comprehensively.