# Circumscribing Circle

## Sum of Areas of Equilateral Triangles Inscribed in Circles

**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.

- Read more about Sum of Areas of Equilateral Triangles Inscribed in Circles
- Log in or register to post comments
- 6033 reads

## Derivation of Formula for Area of Cyclic Quadrilateral

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.

- Read more about Derivation of Formula for Area of Cyclic Quadrilateral
- Log in or register to post comments
- 12293 reads

## The Regular Polygon

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.

- Read more about The Regular Polygon
- Log in or register to post comments
- 6670 reads

## Quadrilateral with one side as diameter of circumscribing circle

**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.

- Read more about Quadrilateral with one side as diameter of circumscribing circle
- Log in or register to post comments
- 12144 reads

## The Cyclic Quadrilateral

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.

$B + D = 180^\circ$

- Read more about The Cyclic Quadrilateral
- Log in or register to post comments
- 4600 reads

## 21 - 24 Solved problems in maxima and minima

**Problem 21**

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

- Read more about 21 - 24 Solved problems in maxima and minima
- Log in or register to post comments
- 87188 reads

## Area of Regular Six-Pointed Star

**Problem**

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

- Read more about Area of Regular Six-Pointed Star
- Log in or register to post comments
- 8413 reads

## Area of Regular Five-Pointed Star

**Problem**

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

- Read more about Area of Regular Five-Pointed Star
- Log in or register to post comments
- 18596 reads

## Derivation of Formula for Radius of Circumcircle

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.

- Read more about Derivation of Formula for Radius of Circumcircle
- Log in or register to post comments
- 49103 reads

## Centers of a 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.

- Read more about Centers of a Triangle
- Log in or register to post comments
- 12748 reads