Look at the triangles QRT and PTS.
These two triangles are similar on the three equal angles1. the angles at T are vertical.
2. Corners PST=RQT as opposite corners.
3. The angles SPT=QRT as opposite corners.
Now calculate the coefficient of similarity.
PS/QR=1:1. That is, the coefficient of similarity =1In other words, the triangles QRT and PTS are equal.
And therefore ST=TQ and PT=TR.
