1. ## Is a quadrilateral cyclic?

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...

2. Hello, logan6!

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 $\displaystyle ADCB.$

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

Hence, quadrilateral $\displaystyle ABCD$ is inscribed in semicircle $\displaystyle ADCB.$

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

3. Thank you very much! You made my bulb glow!

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

Then right triangle is inscribed in circle ACB. (r=AB/2, center in the middle of hypotenuse AB)

Then right triangle is inscribed in circle ADB.
(r=AB/2, center in the middle of hypotenuse AB))

Hence, quadrilateral is inscribed in circle