• Mar 20th 2010, 03:16 PM
logan6
Prove that quadrilateral ABCD is cyclic if diagonal AC is perpendicular to side BC and diagonal BD is perpendicular to side AD.

Need a hint to start...I'm stuck with this...

• Mar 20th 2010, 04:38 PM
Soroban
Hello, logan6!

Quote:

Prove that quadrilateral ABCD is cyclic
if diagonal \$\displaystyle AC \perp BC\$ and diagonal \$\displaystyle BD \perp AD.\$

Code:

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

\$\displaystyle AC \perp BC\$
Then right triangle \$\displaystyle ACB\$ is inscribed in semicircle \$\displaystyle ADCB.\$

Then right triangle \$\displaystyle ADB\$ is inscribed in semicircle \$\displaystyle ADC B.\$

Therefore: .quadrilateral \$\displaystyle ABCD\$ is cyclic.

• Mar 21st 2010, 01:01 AM
logan6
Thank you very much! You made my bulb glow!

Note: I made a petite modification. I hope you don't mind.

http://www.mathhelpforum.com/math-he...692b9e4e-1.gif
Then right triangle http://www.mathhelpforum.com/math-he...5f587e85-1.gif is inscribed in circle ACB. (r=AB/2, center in the middle of hypotenuse AB)

http://www.mathhelpforum.com/math-he...01a1a70f-1.gif
Then right triangle http://www.mathhelpforum.com/math-he...02abc75b-1.gif is inscribed in circle ADB.
(r=AB/2, center in the middle of hypotenuse AB))

Hence, quadrilateral http://www.mathhelpforum.com/math-he...84872ca7-1.gif is inscribed in circle http://www.mathhelpforum.com/math-he...ef48d32e-1.gif