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Two sides and a median bisecting the thi...

Two sides and a median bisecting the third side are respectively proportional to the two sides and corresponding median of other triangle. Prove that the triangle are similar.

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`triangleABC and trianglePQR` in which AD and PS are the medians such that.
`(AB)/(PQ)=(BC)/(QR)=(AD)/(PS)`
To prove : `triangleABC~trianglePQR`
Proof: since, `(AB)/(PQ)=(BC)/(QR)=(AD)/(PS)`
`Rightarrow (AB)/(PQ)=(2BD)/(2QS)=(AD)/(PS)` ( AD and PS are the medians)
`Rightarrow (AB)/(PQ)=(BD)/(QS)=(AD)/(PS)`
`triangleABD~trianglePQS` (SSS criterion of similarity)
`angleB=angleQ` ( corresponding `angles` of similar triangles are equal)
Now in `triangleABC and trianglePQR`
`(AB)/(PQ)=(BC)/(QR)` (given)
and `angleB=angleQ`
`triangleABC~trianglePQR` (SAS criterion of similarity) Hence proved.
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