# Isosceles trapezoid

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An isosceles trapezoid (isosceles trapezium in British English) is a quadrilateral with a line of symmetry bisecting one pair of opposite sides, making it automatically a trapezoid. Two opposite sides (bases) are parallel, the two other sides (legs) are of equal length. The diagonals are of equal length. An isosceles trapezoid's base angles are congruent. Any quadrilateral with one axis of symmetry must be either an isosceles trapezoid or a kite.

## Equal Segments of diagonals

The diagonals of an isosceles trapezoid are equal in length and divide each other into equal segments: $AE=DE$ and $BE=CE$

Therefore the diangle length is

$p=q=\sqrt{ab+c^2}$

## Equal Angles

An isosceles trapezoid has two pairs of equal interior angles. Angles ABC and DCB are equal obtuse angles. Angles BAD and CDA are equal acute angles.

## Area

The area of an isosceles (or any trapezoid) is equal to the average of the bases times the height. In the diagram to the right, $b_1$ = segment $AD$ , $b_2$ = segment $BC$ and $h$ is the length of a line segment between AD and BC and perpendicular to them. The area is given as follows

$A=\frac{h(b_1+b_2)}{2}=\frac{ab+c^2}{2}=\sqrt{\frac{(ab+c^2)^2}{4}-\frac{(b^2-a^2)^2}{16}}=\frac{(a+b)\sqrt{(2c-b+a)(2c+b-a)}}{4}$

Another equivalent formula for the area, which better resembles Heron's formula is:

$A=\frac{a+c}{a-c}\sqrt{(s-c)(s-a)(s-c-b)^2}$

$A=\frac{a+c}{a-c}\sqrt{\frac{(P-2c)(P-2a)(P-2c-2b)^2}{16}}$

where $s=\frac{a+b}{2}+c$ is the semiperimeter of the trapezoid.

\begin{align} R&=\frac{c(a+b)(a-b)}{2}\sqrt{\frac{ab+c^2}{(a^2b^2+c^4)(a-b)^2-2abc^2(a^2+b^2)}}\\ &=c(a+b)\sqrt{\frac{ab+c^2}{4c^2(a+b)^2+2a^2b^2-a^4-b^4}} \end{align}
$r=\dfrac{ab+c^2}{a+b+2c}=\frac{a+b}{2}\sqrt{\frac{(2c-b+a)(2c+b-a)}{(a+b+2c)^2}}$