# Isosceles Trapezoid Proof

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• Sep 4th 2012, 09:35 AM
Senators88
Isosceles Trapezoid Proof
An isosceles trapezoid has points A,B,C, and D where AD and BC are parallel. Prove that any isosceles trapezoid can be inscribed in a circle. Do this by finding a unique point 0 which is equidistant from points A,B,C, and D. Write a proof on how to construct this circle.

Note: Sides AB, BC, and CD have length 1 and side AD has length square root of 3.
• Sep 6th 2012, 08:45 AM
kalyanram
Re: Isosceles Trapezoid Proof
Draw a perpendicular bisector of the sides \$\displaystyle AD\$ and \$\displaystyle DC\$ mark their point of intersection as \$\displaystyle O\$. Now \$\displaystyle O\$ is equidistant from \$\displaystyle A,D\$ and \$\displaystyle C\$. With \$\displaystyle O\$ as the center and radius \$\displaystyle OA\$ (\$\displaystyle =OD= OC \$) draw a circle this circle passes through \$\displaystyle A,D\$ and \$\displaystyle C\$. This will also pass through \$\displaystyle B\$. As quadrilateral ABCD is cyclic.
\$\displaystyle Q.E.D\$.

(Refer to the figure attached).

Kalyan.