Isosceles Right Triangles With a Common Vertex

Isosceles Right Triangles
with a Common Vertex at the Right Angle

Consider isosceles triangles AXB and CXD with right angles at X.

Construct segments AD and BC

Show that these two segments are always congruent.

Show that the lines along which these segments lie are always perpendicular.

Note that the segments AD and BC do not always intersect. For example:

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Isosceles Right Triangles with a Common Vertex at the Right Angle

Isosceles Right Triangles
with a Common Vertex at the Right Angle