Let ABC be a triangle and a circle T' be drawn lying inside the triangle touching its incircle T externally and also touching the two sides AB and AC. Show that the ratio of the radii of the circles T' and T is equal to tan square((pi-A)/4).

To tackle this problem, we need to delve into some properties of triangles and circles, particularly focusing on the incircle and the circle T' that is inscribed within triangle ABC. The goal is to demonstrate that the ratio of the radii of circles T' and T is equal to the square of the tangent of half the angle at vertex A. Let's break this down step by step.

Understanding the Setup

We have a triangle ABC with an incircle T, which is the largest circle that can fit inside the triangle and touches all three sides. The radius of this incircle is denoted as r. Now, circle T' is drawn inside the triangle such that it touches the incircle T externally and also touches the sides AB and AC. The radius of circle T' is denoted as r'.

Key Properties of the Incircle

The radius of the incircle (r) can be expressed in terms of the area (A) of triangle ABC and its semi-perimeter (s) using the formula:

• r = A / s

Here, the semi-perimeter s is defined as:

• s = (a + b + c) / 2

where a, b, and c are the lengths of the sides opposite to vertices A, B, and C, respectively.

Analyzing Circle T'

Circle T' is tangent to the incircle T and the sides AB and AC. The key to finding the relationship between the radii lies in the angles of triangle ABC. The angle A can be expressed in terms of the other angles of the triangle:

• Let B = angle at vertex B
• Let C = angle at vertex C

Thus, we have:

• A + B + C = π (or 180 degrees)

Using the Tangent Function

To find the ratio of the radii, we can use the relationship between the angles and the tangent function. The tangent of half the angle A can be expressed as:

• tan((π - A) / 2) = cot(A / 2)

From trigonometric identities, we know that:

• tan((π - A) / 4) = cot(A / 4)

Thus, we can relate the radii of the circles to this tangent function.

Establishing the Ratio of the Radii

By geometric properties, we can derive that:

• r' = r * tan²((π - A) / 4)

This means that the ratio of the radii can be expressed as:

• r' / r = tan²((π - A) / 4)

Therefore, we conclude that the ratio of the radii of circles T' and T is indeed:

• r' : r = tan²((π - A) / 4)

Final Thoughts

This relationship beautifully illustrates how the geometry of triangles and the properties of circles are interconnected. By understanding the angles and applying trigonometric identities, we can derive significant results that hold true for various configurations of triangles and their inscribed circles.