Answer & Explanation
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 , then . The proof of this is quite simple; just apply the sine law to triangles ABD and CAD.
So for this problem, the ratios is clearly equal to . So is equal to . 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.
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.
So we have