Facts About Cyclic Quadrilateral

Cyclic quadrilateral is a special kind of quadrilateral inscribed in a circle. Inscribe means that the vertices of quadrilateral are on the circumference of the circle. A square, rectangle, isosceles trapezoid are some of cyclic quadrilaterals. Below are the some facts about this special figure.

Cyclic Quadrilateral

1. The opposite angle of cyclic quadrilateral is supplementary.



2. Ptolemy’s Theorem. This theorem is a special case of Ptolemy’s Inequality. He formulated the following equation relating the sides and diagonals.


3. Diagonals as Cords. Using the basic theorem in geometry, we can relate the diagonals with respect to their intersections as follows.


4. Brahmagupta’s Formula. Brahmagupta of India derived a formula to find the area of cyclic quadrilateral.


Where S is the semiperimeter


Use the facts above to solve for the following problems.

Problem 1:  In a cyclic quadrilateral ABCD, If \angle{A}=105°. Find the measure of \angle{C}


Using the first fact, opposite angles are supplementary.





Problem 2: In cyclic quadrilateral ABCD, \angle{C}=\angle{A}=90°. If \overline{BD}=5, and two opposite sides have length 2 and 3. What is the area of quadrilateral?



Draft the quadrilateral as follows,

cyclic quadrilateralSince \Delta BCD and \Delta ABD are right triangle and \overline{BD} is the hypotenuse of both triangles, we can find sides \overline{AD}  and   \overline{BC}  by Pythagorean Theorem.









The area can be found using Brahmagupta’s formula or using the fact that both triangles are right triangle.

Area_{quadrilateral}=Area_{\Delta BCD}+Area_{\Delta ABD}

Area_{quadrilateral}=\displaystyle\frac{\overline{BC}\cdot \overline{CD}}{2}+\displaystyle\frac{\overline{AB}\cdot \overline{AD}}{2}

Area_{quadrilateral}=\displaystyle\frac{1}{2}(3\cdot 4+2\cdot \sqrt{21})

Area_{quadrilateral}=6+\sqrt{21} sq. units




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