Solution of Triangles: Circles Connected with a Triangle - Escribed Circles

When studying the solution of triangles, one important aspect is understanding the circles that can be associated with a triangle. These include the inscribed circle (incircle) and the escribed circles (excircles). In this content, we will focus on the escribed circles, which are circles tangent to one side of the triangle and the extensions of the other two sides.

Escribed Circles

An escribed circle, or excircle, is a circle that lies outside the triangle and is tangent to one of the sides and the extensions of the other two sides. Each triangle has three escribed circles, each tangent to one of the triangle's sides.

Properties of Escribed Circles

• Each excircle is tangent to one side of the triangle and the extensions of the other two sides.
• The center of an excircle is the intersection of two external angle bisectors and the internal bisector of the remaining angle.
• The radius of an excircle is called the exradius.

Formulas Related to Escribed Circles

The radius of an excircle, denoted as ( r_a, r_b, ) or ( r_c ), depending on which side of the triangle it is tangent to, can be found using the formula:

[ r_a = \frac{K}{s - a} ] [ r_b = \frac{K}{s - b} ] [ r_c = \frac{K}{s - c} ]

where ( K ) is the area of the triangle, ( s ) is the semiperimeter, and ( a, b, c ) are the lengths of the sides of the triangle.

The semiperimeter ( s ) is given by:

[ s = \frac{a + b + c}{2} ]

The area ( K ) can be found using Heron's formula:

[ K = \sqrt{s(s - a)(s - b)(s - c)} ]

Differences and Important Points

Aspect	Incircle	Excircle
Definition	A circle inscribed in the triangle	A circle escribed about the triangle
Tangency	Tangent to all three sides	Tangent to one side and extensions of the other two sides
Center	Intersection of angle bisectors	Intersection of two external angle bisectors and one internal angle bisector
Radius notation	( r )	( r_a, r_b, r_c )
Formula for radius	( r = \frac{K}{s} )	( r_a = \frac{K}{s - a} ) (and similar for ( r_b, r_c ))

Examples

Example 1: Finding the Radius of an Excircle

Given a triangle with sides ( a = 7 ), ( b = 9 ), and ( c = 12 ), find the radius of the excircle opposite to side ( a ).

Solution:

First, calculate the semiperimeter ( s ):

[ s = \frac{a + b + c}{2} = \frac{7 + 9 + 12}{2} = 14 ]

Next, use Heron's formula to find the area ( K ):

[ K = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{14(14 - 7)(14 - 9)(14 - 12)} = \sqrt{14 \cdot 7 \cdot 5 \cdot 2} = 14\sqrt{10} ]

Now, find the radius of the excircle opposite to side ( a ) using the formula:

[ r_a = \frac{K}{s - a} = \frac{14\sqrt{10}}{14 - 7} = 2\sqrt{10} ]

Example 2: Comparing Excircle Radii

Using the same triangle from Example 1, compare the radii of the excircles opposite to sides ( a ) and ( b ).

Solution:

We already found ( r_a = 2\sqrt{10} ) in Example 1.

Now, calculate ( r_b ):

[ r_b = \frac{K}{s - b} = \frac{14\sqrt{10}}{14 - 9} = \frac{14\sqrt{10}}{5} ]

Since ( 2\sqrt{10} < \frac{14\sqrt{10}}{5} ), we can conclude that the excircle opposite to side ( b ) has a larger radius than the excircle opposite to side ( a ).

Understanding the properties and formulas related to escribed circles is crucial for solving problems in triangle geometry, especially in the context of competitive exams where time efficiency and accuracy are key.