Question

Full steps and all reasons, thanks!​

Answer 1

You can focus on the two sub-triangles ABC and ACD

We know that they are both right triangles, because we're given that AC and BD are perpendicular.

The sides AB and AD are the hypothenuses of these triangles, so we know their length through the pythagorean theorem:

$AD^2 = AC^2+CD^2$

$AB^2 = AC^2+BC^2$

But since we're given $BC=CD$, the expressions for AD and AB are actually the same expression. This proves that AD=AB, and thus that the triangle is isosceles.