in triangle pqr ps is the median through p and x is the midpoint of ps. qx is produced to meet pr at y. prove that py =1/3 pr
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In ∆PTQ & ∆STR
PT=ST ( By construction)(1)
∠PTQ = ∠STR (vertically opposite angles)
QT=RT (PT is a median, T is a midpoint of QR)
∆PTQ≅∆STR (By SAS congruence rule)
PQ= SR (By CPCT)(2)
In ∆PSR
PR +SR> PS
[Sum of any two sides of a triangle is greater than the third side]
PR +SR> PT + TS (PS= PT+TS)
PR + PQ >PT + TS (from eq. 2)
PR + PQ >PT + PT (from eq. 1)
PR + PQ > 2PT
Hence, the sum of any two sides of a triangle is greater than twice the median with respect to the third side.
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