# Prove that trapezoid inscribed in a circle is isosceles.

• Dec 26th 2009, 01:49 AM
winsome
Prove that trapezoid inscribed in a circle is isosceles.
Prove that trapezoid inscribed in a circle is isosceles?
I have a trapezoid ABCD ...dont know how to prove digonals AC = BD? Help...
• Dec 26th 2009, 06:05 AM
HallsofIvy
Quote:

Originally Posted by winsome
Prove that trapezoid inscribed in a circle is isosceles?
I have a trapezoid ABCD ...dont know how to prove digonals AC = BD? Help...

Draw lines from the center of thecircle to each vertex. consider the triangles formed and use the fact that the sides from the center to the vertices are all congruent. You will probably need to consider two cases: where the center of the circle is inside the trapezoid and where it isn't.
• Dec 26th 2009, 07:03 AM
Soroban
Hello, winsome!

Is it really this simple?

Quote:

Prove that trapezoid inscribed in a circle is isosceles.
Code:

              * * *         A o-----------o B         */            \*       */              \*       /                \     D o-------------------o C       *                  *       *                  *       *                *         *              *           *          *               * * *

$\displaystyle \begin{array}{cccc}AB \parallel CD && \text{D{e}finition of trapezoid} \\ \\ \text{arc}(AD) \,=\,\text{arc}(BC) && \text{Parallel lines intercept equal arcs} \\ \\ AD \:=\:BC &&\text{Equal arcs subtend equal chords.} \\ \\ \therefore \:ABCD\text{ is isosceles.} \end{array}$

• Dec 27th 2009, 07:57 PM
winsome
Quote:

Originally Posted by Soroban
Hello, winsome!

Is it really this simple?

Code:

              * * *         A o-----------o B         */            \*       */              \*       /                \     D o-------------------o C       *                  *       *                  *       *                *         *              *           *          *               * * *

$\displaystyle \begin{array}{cccc}AB \parallel CD && \text{D{e}finition of trapezoid} \\ \\ \text{arc}(AD) \,=\,\text{arc}(BC) && \text{Parallel lines intercept equal arcs} \\ \\ AD \:=\:BC &&\text{Equal arcs subtend equal chords.} \\ \\ \therefore \:ABCD\text{ is isosceles.} \end{array}$

Thanx for Reply but i have few more issues,
is there another solution if we not consider
the property of Parallel lines intercept equal arcs ?as i need the solution without using this property.