In △ A B C, side BC is divided by D in a ratio of 5 to 2 and BA is divided by E in a ratio of 3 to 4 as shown in Figure. Find the ration in which F divides the cevians AD and CE i.e. find E F : F C and D F : F A.

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In   △  A  B  C, side BC is divided by D in a ratio of 5 to 2 and BA is divided by E in a ratio of 3 to 4 as shown in Figure. Find the ration in which F divides the cevians AD and CE i.e. find   E  F  :  F  C and   D  F  :  F  A.

yermarvg · Accepted Answer

Step 1A general method to solving these kinds of things is through the Ratio Lemma, which is often applied in high school olympiad geometry. It states that if AD is a cevian in   △  A  B  C, then                     B        D                    D        C              =                    A        B                    A        C              ⋅                    sin        ⁡        ∠        B        A        D                    sin        ⁡        ∠        C        A        D            . The proof of this is quite simple; just apply the sine law to triangles ABD and CAD.Step 2So for this problem, the ratios   sin  ⁡  ∠  B  A  D  :  sin  ⁡  ∠  C  A  D is clearly equal to   sin  ⁡  ∠  E  A  F  :  sin  ⁡  ∠  C  A  F. So   E  F  :  F  C is equal to   (  B  D      /    D  C  )  (  A  E      /    A  B  )  =  10      /    7.   D  F  :  F  A is calculated similarly.The Ratio Lemma can also be used to prove many basic facts in projective geometry; e.g. the invariance of the cross-ratio when projected through a point onto a line.

Levi Rasmussen · Accepted Answer

Step 1Use mass points. We assign B a mass of 4. So A has a mass of 3 and C has a mass of 10. E has a mass of 7 and D has a mass of 14.Step 2So we have             A      F              F      D        =      14    3  then             E      F              F      C        =      10    7

In triangle ABC, side BC is divided by D in a ratio of 5 to 2 and BA is divided by E in a ratio of 3 to 4 as shown in Figure. Find the ration in which F divides the cevians AD and CE i.e. find EF:FC and DF:FA.

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