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.

Ishaan Booker

Ishaan Booker

Answered question

2022-07-16

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.

Answer & Explanation

yermarvg

yermarvg

Beginner2022-07-17Added 19 answers

Step 1
A 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 2
So 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

Levi Rasmussen

Beginner2022-07-18Added 6 answers

Step 1
Use 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 2
So we have A F F D = 14 3
then E F F C = 10 7

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