The figure below shows
a
tangential or circumscribed quadrilateral ABCD. A_{1}, B_{1}, C_{1}, and D_{1} are
the
midpoints of AB, BC, CD, and AD, respectively. A_{1}A_{2}, B_{1}B_{2}, C_{1}C_{2}, and
D_{1}D_{2} are
perpendicular to CD, AD, AB, and BC, respectively. A_{1}A_{2} cuts B_{1}B_{2}
at A_{3}, B_{1}B_{2} cuts C_{1}C_{2} at B_{3}, C_{1}C_{2} cuts D_{1}D_{2} at C_{3}, and D_{1}D_{2} cuts A_{1}A_{2} at
D_{3}. Prove that A_{3}B_{3}C_{3}D_{3} is a tangential quadrilateral.

See also:

Art of
Problem 1351 using iPad Pro Apps.