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प्रश्न
In a quadrilateral ABCD, P is an internal point. Prove that AP + BP + CP + DP > Semi-perimeter of quadrilateral ABCD.
सिद्धांत
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उत्तर
ABCD is a quadrilateral, P is an internal point.
We know that, The sum of two sides of a triangle is greater than the third side.
In ΔDPC, DP + CP > DC ...(1)
In ΔAPB, AP + BP > AB ...(2)
In ΔCPB, CP + BP > CB ...(3)
In ΔAPD, DP + AP > AD ...(4)
Adding equation (1), (2), (3) and (4),
DP + CP + AP + BP + CP + BP + DP + AP > DC + AB + CB + AD
2(AP + BP + CP + DP) > AB + CB + DC + AD
2(AP + BP + CP + DP) > Semi-perimeter of quadrilateral ABCD
Hence, proved.
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