In the given figure $PQRS$ is a cyclic quadrilateral. $PQ$ and $SR$ are produced up to center $O$ of the circle. $OT$ and $OR$ are the radii of the circle. $QR$ and $PS$ are produced upto the point $V$ and $PV$ and $TR$ are produced up to the point $U$. Prove that: $RV=UV$.
My attempt:
Given: $PQRS$ is a cyclic quadrilateral
To Prove: $RV=UV$
Proof
- $$\angle PSR =\angle URS+\angle SUR$$
Since $\triangle{OTR}$ is an isosceles triangle with $OT=OR$, $$\angle{OTR}=\angle{ORT}=\angle{ORQ}+\angle{QRT},$$ i.e. $$\angle{URV}=\angle{QRT}=\angle{OTR}-\angle{ORQ}\tag1$$
Also, using $\angle{QPS}=\angle{ORQ}$, $$\begin{align}\angle{RUV}&=\angle{OTR}-\angle{QPS}\\&=\angle{OTR}-\angle{ORQ}\end{align}\tag2$$
It follows from $(1)(2)$ that $\angle{URV}=\angle{RUV}$.