Problem 18 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Prove that the distance between the circum-centre ( \(O\) ) and the in-center \((I)\) is \(O I=R \times \sqrt{\left(1-8 \sin \left(\frac{A}{2}\right) \sin \left(\frac{B}{2}\right) \sin \left(\frac{C}{2}\right)\right)}\)

Step-by-Step Solution

Answer

The proof involves using the properties of a triangle's in-center and circum-center along with the sine and cosine rules, and ends with the expression for \(OI = R \times \sqrt{1-8 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}}\)

Step 1: Understand the Properties

First, remember the properties of in-center (denoted as \(I\)) and circum-center (denoted as \(O\)) of a triangle. The in-center is the center of the inscribed circle (incircle) of the triangle, it can be determined as the intersection of the angle bisectors of the triangle. The circum-center is the center of the circumscribed circle (circumcircle) of the triangle, and for any triangle's circum-center, the distances between the circum-center and the vertices of the triangle are equal.

Step 2: Derive the Formula for Required Distance

Knowing that radius of the incircle is given by \(r = \frac{abc}{4R}\), where \(a, b, c\) are sides of the triangle and \(R\) is the circum-radius. Also, using cosine rule, \(\cos A = \( \frac{b^2 + c^2 - a^2}{2bc} \) and then substituting for \(r\) and \(R\), the radius of the circle can be related back to the sides and angles of the triangle. Then by utilising the sine rule, \(a = 2RsinA\), \(b = 2RsinB\) and \(c = 2RsinC\), we can evaluate \(R\).

Step 3: Calculate Numerical Values

You will implement the previous step for the given triangle with specified angles and sides, \(A, B, C\) and replace the lengths \(a, b, c\) with their respective evaluated formulas. Calculate the numerical value of the expression inside the square root.

Step 4: Substitute Back in the Main Formula

The required distance \(OI\) is calculated by substituting the respective value found earlier multiplied by \(R\) inside the square root.

Key Concepts

Circumcenter

The circumcenter of a triangle is one of the key points of the triangle and holds important properties. It's the center of the circumcircle, which is the circle that passes through all three vertices of the triangle.
The circumcenter is equidistant from all three vertices, meaning the distances from the circumcenter to each vertex are equal. This property is essential for certain trigonometric proofs, as it helps establish relationships between different components of the triangle.
In geometric terms, the circumcenter is found as the point of intersection of the perpendicular bisectors of the sides of the triangle. Depending on whether the triangle is acute, right, or obtuse, the circumcenter can lie inside, on, or outside the triangle, respectively.

Incenter

The incenter of a triangle is another fundamental point that is often used alongside the circumcenter in geometry and trigonometry. The incenter is the point where the angle bisectors of a triangle meet, and it is the center of the incircle, which is the largest circle that fits inside the triangle.
There are unique properties of the incenter to keep in mind:

• The incenter is equidistant from all three sides of the triangle, which makes the radius of the incircle uniform.
• The formula for the inradius, or the radius of the incircle, is given by \( r = \frac{A}{s} \), where \( A \) is the area of the triangle and \( s \) is the semi-perimeter.

The position of the incenter is always inside the triangle, irrespective of the type of triangle, providing a stable center point for internal problems.

Triangle Properties

Triangles are a fundamental geometric shape with several basic properties that are used in many problems and proofs. Here are a few of their key properties:

• The sum of the three interior angles of a triangle is always \(180^\circ\).
• The area \(A\) can be calculated using different formulas, such as \( \frac{1}{2} \times \, base \, \times \, height \) or using Heron's formula.
• The perimeter is the sum of its three side lengths: \(a+b+c\).

In trigonometry, the relationships between the angles and lengths are used to establish more advanced formulas, such as \cosine and sine rules, to connect different parts of triangle geometry to each other.

Sine Rule

The sine rule, also known as the law of sines, is a crucial equation in trigonometry that relates the lengths of sides of a triangle to the sines of its angles. It's given by:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \]where \( a, b, c \) are the lengths of the sides opposite angles \( A, B, \) and \( C \) respectively, and \( R \) is the circumradius.
This rule is especially useful when dealing with non-right triangles, as it provides a way to solve for unknown sides or angles when given only partial information.
In combination with other trigonometric identities and rules, the sine rule helps in constructing proofs and solving complex geometric problems, such as finding relationships between triangle centers and the triangle's sides and angles.