Hello, murphmath!

Given a quadrilateral $\displaystyle ABCD$ with diagonals $\displaystyle AC$ and $\displaystyle BD$ intersecting at point $\displaystyle E$,

with the following givens: $\displaystyle BC = CD$ and $\displaystyle AB = AD.$

Prove that $\displaystyle AC$ is perpendicular to $\displaystyle BD.$ We have a kite-shaped figure . . . Code:

A
*
* | *
* | *
D * - - + - - * B
* |E *
* | *
* | *
* | *
*|*
*
C

We are given: .$\displaystyle AB = AD,\;BC = CD$

Point $\displaystyle A$ is equidistant from $\displaystyle B$ and $\displaystyle D.$

Point $\displaystyle C$ is equidistant from $\displaystyle B$ and $\displaystyle D.$

Hence, $\displaystyle AC$ is the perpendicular bisector of $\displaystyle BD.$

. . Therefore: .$\displaystyle AC \perp BD$