Question

ABCD is quadilateral in which AD = BC and $\angle$ ADC = $\angle$BCD
Show that A B C and D are concyclic.

Answer 1

It can be proved that ABCD is an Isosceles Trapezium, as follows:
Produce AD and BC to meet at E.
Now $\angle$EDC = $\angle$ECD (supplementary angles of equal $\angle$ADC and $\angle$BCD).
Therefore triangle EAB and also triangle EDC are isosceles triangles.
EC = ED and AD = BC implies that DC intersects sides AE and BE in equal proportion. From the proportionate theorem of triagles, it follows that DC is parallel to base AB and $\frac{DC}{AB}=\frac{ED}{EA}$.
Now in ABCD, AD = BC and CD is parallel to AB.
This shows ABCD is an Isosceles Trapezium, which is always con-cylic. Hence ABCD is concylic. ------ proved.