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.