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draw perpendicular XZ from X on PQ
since XP = XQ
∆ XPQ is an isosceles triangle.
in isosceles triangle, perpendicular from vertex bisects the base
thus, ZP = ZQ
ZP + PY = ZQ + PY
ZP + PY = ZQ + QR. ( PY = QR)
ZY = ZR
now in ∆ XYR
perpendicular XZ bisects YR
hence, ∆ XYR is an isosceles ∆
so XY = XR
proved
(note: we can also prove last part by proving triangles XZY and XZR congruent)
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