Engineering Mathematics
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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.
 

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
 

$A = \sqrt{(s - a)(s - b)(s - c)(s - d)}$

 

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

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.
 

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.
 

Cyclic quadrilateral inscribed in a circle of unknown radius

 

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.

$A + C = 180^\circ$

$B + D = 180^\circ$

 

Cyclic quadrilateral

 

21 - 24 Solved problems in maxima and minima

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

Area of Regular Six-Pointed Star

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

Area of Regular Five-Pointed Star

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

Derivation of Formula for Radius of Circumcircle

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

$R = \dfrac{abc}{4A_t}$

where At is the area of the inscribed triangle.
 

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.
 

incenter-incircle.jpg

 

The radius of incircle is given by the formula

$r = \dfrac{A_t}{s}$

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