Problem 246 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Let \(A B C\) be a triangle with incentre \(I\) and inradius \(r\). Let \(D, E, F\) be the feet of the perpendiculars from \(I\) to the sides \(B C, C A\) and \(A B\) respectively. If \(r_{1}, r_{2}\) and \(r_{3}\) are the radii of circles inscribed in the quadrilateral \(A F I E, B D I F\) and \(C E I D\) respectively, prove that \(\frac{r_{1}}{r-r_{1}}+\frac{r_{2}}{r-r_{2}}+\frac{r_{3}}{r-r_{3}}=\frac{r_{1} r_{2} r_{3}}{\left(r-r_{1}\right)\left(r-r_{2}\right)\left(r-r_{3}\right)} .\)

Step-by-Step Solution

Answer

Following the steps above, the equation simplifies into the desired result \(\frac{r_{1}}{r-r_{1}}+\frac{r_{2}}{r-r_{2}}+\frac{r_{3}}{r-r_{3}}=\frac{r_{1} r_{2} r_{3}}{\left(r-r_{1}\right)\left(r-r_{2}\right)\left(r-r_{3}\right)}\). As per the equation, the sum of the ratios of the inradii of the quadrilaterals to the difference of the inradius of the triangle and the respective inradii of the quadrilaterals equals the ratio of the product of the inradii of quadrilaterals to the product of the differences of the inradius of the triangle and the respective inradii of the quadrilaterals. Hence, proved.

Step 1: Formation of Triangles and Quadrilaterals

Draw the triangle \(ABC\) such that side \(BC\) lies on the x-axis. Mark the centre of the triangle, \(I\) and draw the inradius, \(r\). Then draw perpendiculars from \(I\) to sides \(BC\), \(CA\) and \(AB\), and mark their feet as \(D\), \(E\) and \(F\) respectively.

Step 2: Radii of Circles Inscribed in Quadrilaterals

Inscribe circles in the quadrilaterals \(AFIE\), \(BDIF\) and \(CEID\) with radii \(r_1\), \(r_2\) and \(r_3\) respectively. These radii can be seen as the distances from the incentre \(I\) to the sides of the triangle \(ABC\).

Step 3: Area of Triangle

The area of triangle \(ABC\) can be written as two times the sum of area of quadrilaterals \(AFIE\), \(BDIF\) and \(CEID\), which can be represented as \(2r(r_1 + r_2 + r_3)\).

Step 4: Area of Quadrilaterals

The areas of the three quadrilaterals \(AFIE\), \(BDIF\), \(CEID\) can also be represented in terms of the respective inradii and the distances from the incentre \(I\) to the sides of the original triangle \(ABC\). Now the areas of the quadrilaterals can be expressed as \(r_1(r - r_1)\), \(r_2(r - r_2)\) and \(r_3(r - r_3)\) respectively.

Step 5: Equating the Areas

Setting the two expressions for the area of triangle \(ABC\) equal results in the equation: \(2r(r_1 + r_2 + r_3) = r_1(r - r_1) + r_2(r - r_2) + r_3(r - r_3)\). Simplifying this equation leads to the desired result.

Key Concepts

Inradius

The inradius of a triangle is a fundamental concept in geometry. It refers to the radius of the circle that can be inscribed within a triangle, touching all three sides. This circle is known as the incircle. To find the inradius, denoted by \(r\), you can use the formula:
\[ r = \frac{A}{s} \]
where \(A\) is the area of the triangle, and \(s\) is the semi-perimeter of the triangle (half of the triangle's perimeter).The inradius plays a crucial role in various geometric calculations. It helps in determining the area of a triangle when side lengths are known. For instance, once the inradius is known, the area of the triangle can be conveniently calculated using the formula \( A = r \times s \).Understanding the concept of inradius helps in solving problems involving inscribed circles and related areas.

Incircle

An incircle, or inscribed circle, is the largest circle that fits neatly inside a polygon, such as a triangle, touching each side. For a triangle, the center of the incircle is known as the incentre, which is the point where the angle bisectors of the triangle intersect.

• The incircle's radius is called the inradius.
• The circle is tangent to each of the triangle's sides.
• Its centre, the incentre, is equidistant from all triangle sides.

The incircle is significant in both practical and theoretical geometry. It aids in area calculations and understanding symmetry and properties related to tangent circles. Its usage extends beyond triangles to quadrilaterals and polygons, where different rules apply but the fundamental idea remains: the circle touches each side, making it an essential element in geometry.

Quadrilateral

A quadrilateral is a four-sided polygon with four vertices and four edges. It is fundamental in geometry, serving as the basis for a variety of geometric figures. Quadrilaterals can be simple, like squares and rectangles, or more complex, like kites and trapezoids.In advanced geometry, quadrilaterals may include inscribed circles, similar to incircles in triangles. For example, in the problem setup involving quadrilaterals such as \(AFIE\), \(BDIF\), and \(CEID\), circles are inscribed, and the radii of these circles (\(r_1, r_2, r_3\)) are crucial for solving geometry problems related to areas and other properties.Understanding the properties of different types of quadrilaterals, such as whether they are convex or concave, symmetrical, or regular, helps in analyzing problems and identifying the necessary geometric properties to find solutions.

Triangle Area

The area of a triangle is one of the essential aspects of its geometry. The formula to find the area \(A\) of a triangle with base \(b\) and height \(h\) is:
\[ A = \frac{1}{2} \, b \, h \]
However, if you know the side lengths of the triangle, Heron's formula is another method to find the area. It states: \[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
where \(s\) is the semi-perimeter: \[ s = \frac{a+b+c}{2} \]
The area calculation is directly tied to the inradius of the triangle, as mentioned earlier with the formula \( A = r \times s \). This linkage to other geometric properties allows us to explore triangular relationships further, like equating the area of the triangle to related inscribed quadrilateral areas.Mastering how to calculate a triangle's area empowers solving more significant geometric puzzles and understanding spatial layouts.