Circumscribing Circle

Problem
An equilateral triangle is inscribed within a circle whose diameter is 12 cm. In this triangle a circle is inscribed; and in this circle, another equilateral triangle is inscribed; and so on indefinitely. Find the sum of the areas of all the triangles.
 

For a cyclic quadrilateral with given sides a, b, c, and d, the formula for the area is given by
 

Where s = (a + b + c + d)/2 known as the semi-perimeter.
 

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.
 

Problem PG-010
The quadrilateral ABCD shown in Fig. PG-010 is inscribed in a circle with side AD coinciding with the diameter of the circle. if sides AB, BC, and CD are 8 cm, 10 cm, and 12 cm long, respectively, find the radius of the circumscribing circle.
 

A quadrilateral is said to be cyclic if its vertices all lie on a circle. In cyclic quadrilateral, the sum of two opposite angles is 180° (or π radian); in other words, the two opposite angles are supplementary.

Problem 21
Find the rectangle of maximum perimeter inscribed in a given circle.
 

Problem
Find the area of the regular six-pointed star inscribed in a circle of radius 20 cm.
 

Problem
Find the area of the regular five-pointed star inscribed in a circle of radius 20 cm.
 

The formula for the radius of the circle circumscribed about a triangle (circumcircle) is given by
 

where A_t is the area of the inscribed triangle.
 

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 A_t = area of the triangle and s = ½ (a + b + c). See the derivation of formula for radius of incircle.