Three circles, the radii of which are equal to 2, 3 and 4, in pairs are tangent externally.

Three circles, the radii of which are equal to 2, 3 and 4, in pairs are tangent externally. Find the area of a triangle whose vertices are the centers of the circles.

The figure shows that the sides of the triangle ABC are equal to the sums of the radii of the contiguous circles. AB = AA1 + A1B = 4 + 3 = 7 cm, AC = AC1 + CC1 = 4 + 2 = 6 cm, CB = BB1 + CB1 = 3 + 2 = 5 cm.

We define the area of a triangle by Heron’s theorem.

Sас = √р * (р – а) * (р – b) * (p – c), where р – half-perimeter of a triangle, а, b, c – sides of a triangle.

p = (AB + BC + CA) / 2 = (7 + 6 + 5) / 2 = 18/2 = 9 cm.

Sav = √9 * (9 – 7) * (9 – 6) * (9 – 5) = √9 * 2 * 3 * 4 = √216 = 6 * √6 cm2.

Answer: The area of the triangle is 6 * √6 cm2.