Problem
A circle of radius 1 inch is inscribed in an equilateral triangle. A smaller circle is inscribed at each vertex, tangent to the circle and two sides of the triangle. The process is continued with progressively smaller circles. What is the sum of the circumference of all the circles.
Quadrilateral circumscribing a circle (also called tangential quadrilateral) is a quadrangle whose sides are tangent to a circle inside it.
Area,
Where r = radius of inscribed circle and s = semi-perimeter = (a + b + c + d)/2
Derivation for area
Rhombus is a quadrilateral with all sides equal (equilateral). Rectangle is a quadrilateral with all included angles are equal (equiangular). Square is both equilateral and equiangular, thus square is a regular polygon. Regular polygons are polygons with all sides equal and all included angles equal. Meaning, regular polygons are both equilateral and equiangular.
The radius of incircle is given by the formula
where At = area of the triangle and s = semi-perimeter.
This page will define the following: incenter, circumcenter, orthocenter, centroid, and Euler line.
Incenter
Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle.
The radius of incircle is given by the formula
where At = area of the triangle and s = ½ (a + b + c). See the derivation of formula for radius of incircle.
