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triangle on sides of triangle, equal ratio

Source: Polish MO Finals 1952 p2

August 28, 2024
geometryareasequal ratioratio

Problem Statement

On the sides BC BC , CA CA , AB AB of the triangle ABC ABC , the points M M , N N , P P are taken, respectively, in such a way that BMMC=CNNA=APPB=k,\frac{BM}{MC} = \frac{CN}{NA} = \frac{AP}{PB} = k, where k k means a given number greater than 1 1 , then the segments AM AM , BN BN , CP CP were drawn . Given the area S S of the triangle ABC ABC , calculate the area of the triangle bounded by the lines AM AM , BN BN and CP CP .
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