Problem 4

Let

ABCDE

be a convex pentagon such that

BC=DE

. Assume that there is a point

T

inside

ABCDE

with

TB=TD

TC=TE

and

\angle ABT=\angle TEA

. Let line

AB

intersect lines

CD

and

CT

at points

P

and

Q

, respectively, with

P,B,A,Q

in that order on the line. Let line

AE

intersect lines

CD

and

DT

at points

R

and

S

, respectively, with

R,E,A,S

in that order on the line. Prove that the points

P

S

Q

R

lie on a circle.

Step 5 of 6: Chase angles at

P

using the circle

CDQS

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\angle QPR=\angle QPC=\angle QCD-\angle PQC

Detailed analysis

Since $P,R$ both lie on line $CD$ , $\angle QPR=\angle QPC$ . Because $C,D,Q,S$ are concyclic, the inscribed angle $\angle QCD$ equals $\angle QSD$ ; and the exterior angle of triangle $PQC$ at $C$ gives $\angle QCD=\angle QPC+\angle PQC$ , so $\angle QPR=\angle QPC=\angle QCD-\angle PQC$ .