A circle is inscribed in triangle ABC with sides a, b, c. Tangents to the circle parallel to the sides of the triangle are constructed. Each of these tangents cuts off a triangle from ∆ABC. In each of these triangles, a circle is inscribed.
Find the sum of the areas of all four inscribed circles (in terms of a, b, c).
To find the sum of the areas of all four inscribed circles, we need to break down the problem into smaller steps.
Step 1: Find the radius of the incircle of triangle ABC.
The radius of the incircle, denoted by r, can be found using the formula:
r = (Area of triangle ABC)/(Semi-perimeter of triangle ABC)
The Area of triangle ABC can be found using Heron's formula:
Area = sqrt(s * (s - a) * (s - b) * (s - c))
where s is the semi-perimeter of triangle ABC, given by:
s = (a + b + c) / 2
Step 2: Find the inradius of the smaller triangles.
The inradius of the smaller triangles can be found using the same formula as in Step 1, with the corresponding sides of each smaller triangle. Let's denote the radii of the four smaller circles as r1, r2, r3, and r4.
Step 3: Calculate the areas of the four smaller circles.
The area of a circle can be found using the formula:
Area = π * (radius)^2
So, the areas of the four smaller circles are:
Area1 = π * r1^2
Area2 = π * r2^2
Area3 = π * r3^2
Area4 = π * r4^2
Step 4: Find the sum of the areas of the four smaller circles.
Sum of areas = Area1 + Area2 + Area3 + Area4
Now, let's put it all together:
Step 1: Calculate the radius of the incircle of triangle ABC:
r = sqrt(s * (s - a) * (s - b) * (s - c)) / s
Step 2: Calculate the inradius of the smaller triangles:
r1 = r * (a / (a + b + c))
r2 = r * (b / (a + b + c))
r3 = r * (c / (a + b + c))
r4 = r * (a / (a + b + c))
Step 3: Calculate the areas of the four smaller circles:
Area1 = π * r1^2
Area2 = π * r2^2
Area3 = π * r3^2
Area4 = π * r4^2
Step 4: Find the sum of the areas of the four smaller circles:
Sum of areas = Area1 + Area2 + Area3 + Area4 = π * (r1^2 + r2^2 + r3^2 + r4^2)
So, the sum of the areas of the four inscribed circles is π times the sum of the squares of the inradii of the smaller triangles.