Given: △PQR, QX and RX are bisectors of ∠Q and ∠R and XT⊥PQ and XS⊥QR
Construction: Draw XM⊥PR, Join PX
In △XSR and △XMR
∠XRS=∠XRM (XS bisects ∠R)
XR=XR
∠XSR=∠XMR=90∘
Thus, △XSR≅△XMR (ASA rule)
Hence, XS=XM (BY CPCT)...(1)
Similarly, △XQS≅△XQT
Hence, XS=XT (By CPCT)....(2)
Hence, by (1) and (2)
XM=XT
Now, In △PXT and △PXM
XT=XM (Proved above)
∠XMP=∠XTP (Each 90∘)
PX=PX (Common)
Thus, △PXT≅△PXM (SAS rule)
Hence, ∠TPX=∠MPX (by CPCT)
or PX bisects ∠P