Problem 223 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Three equal circles touch one another. Find the radius of the circle which touches all three. \\{Ans. \(\left.\left(\frac{2}{\sqrt{3}} \pm 1\right) r\right\\}\)

Step-by-Step Solution

Answer

The radius of the fourth circle which touches all three is \( \left.( \frac{2}{\sqrt{3}} \pm 1\right) r \).

Step 1: Draw a Diagram

Draw a diagram having three equal circles with radius \( r \) touching each other, with their centers forming an equilateral triangle. Then, draw another circle, of radius \( R \), that's touching all three circles.

Step 2: Forming Similar Triangles

Noticing that the line connecting the center of the fourth circle with the center of any of the three initial ones bisects the side of the equilateral triangle into two equal segments, it's seen that two similar triangles are formed - one with sides \( r \), \( R+r \), and the other with sides \( r \), \( R-r \).

Step 3: Set up Equations from Ratios of Similar Triangles

From the similar triangles we can set up these ratio equations: \[ \frac {R - r} {r} = \frac {r} {2r + R} \quad and \quad \frac {R + r} {r} = \frac {r} {R - r} \]

Step 4: Solve the Equations

Solving the equations, we get: \[ R = \left.\left(\frac {2}{\sqrt{3}} \pm 1\right) r \right \] which are the radii of the two circles that can touch the three initial circles, one from the inside and the other from the outside.

Key Concepts

Understanding Circles in Geometry

Circles are fundamental shapes in geometry. They are defined as a set of points that are equidistant from a central point. This distance is known as the radius, denoted by \( r \).
When you have multiple circles, interesting relationships can occur. For instance, when three equal circles touch each other, their centers form the vertices of an equilateral triangle.
This is because each circle's radius serves as a side of the triangle, creating symmetry and equal spacing.

• Each pair of circles touches precisely at one point.
• All circles have the same radius, so their centers are equally spaced.

These relationships help in forming geometric constructions, such as determining the radius of another circle that touches all three — a core part of our exercise.

The Role of Similar Triangles

In our context, similar triangles play an essential role in solving the problem. Similar triangles have the same shape but possibly different sizes. This means their corresponding angles are equal, and their sides are proportional.
When the centers of the three circles form an equilateral triangle, adding a fourth circle that touches all three creates configurations that can be analyzed using similar triangles.
Here’s how it works:

• The line connecting the center of the fourth circle with one of the initial circles bisects the equilateral triangle’s side.
• This creates similar triangles, where the sides can be represented as \( r \), \( R+r \), and \( R-r \).

Using the properties of similar triangles lets us set up ratios to find relationships between the circle’s radii. This strategy is crucial for solving for \( R \), the unknown radius of the new circle.

Exploring Equilateral Triangles

An equilateral triangle is a type of polygon where all three sides and angles are equal. It’s a perfect example of symmetry and balance, making it a critical component in many geometric exercises.
In problems involving circles, often these triangles indicate the optimal configuration for space efficiency and even spacing.

• In our specific exercise, the equilateral triangle is formed by the centers of the original circles.
• Each side of this triangle equals the diameter of the circles because the circle's radius extends from the center to the point of contact.

This means each side of the triangle is equal to \( 2r \). The symmetry and properties of equilateral triangles become pivotal when exploring spatial relations in geometry, allowing us to derive insights and solve related problems efficiently, like finding the new circle's radius.