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प्रश्न
In ΔPQR, the bisector of ∠Q and ∠R intersect in point X. Line PX intersects side QR in point Y, then prove that `("PQ" + "PR")/"QR" = "PX"/ "XY"`.
प्रमेय
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उत्तर
Given: In ΔPQR, ray QX and ray RX are bisectors of ∠Q and ∠R respectively.
To prove: `("PQ" + "PR")/"QR" = "PX"/ "XY"`
Proof: In ΔPQY, ray QX is an angle bisector of ∠PQY.
∴ `"PX"/"XY" = "PQ"/"QY"` ...(I) [Angle bisector theorem]
In ΔPRY, ray RX is an angle bisector of ΔPRY.
∴ `"PX"/"XY" = "PR"/"RY"` ...(II) [Angle bisector theorem]
∴ `"PX"/"XY" = "PQ"/"QY" = "PR"/"RY"` ...From (I) and (II)
∴ `"PX"/"XY" = ("PQ" + "PR")/("QY" + "RY")` ...[Theorem on equal ratios]
∴ `"PX"/"XY" = ("PQ" + "PR")/"QR"` ...(Q-Y-R)
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