Question

triangle AABC is given. let AC=b, AB=c, and angle ABC =3ABC. Points D and E lie on line segment AC such that angle ABD = angle DBE=angle EBC. Express the line segments AD, DE, and EC using the given lengths b and c. ​

Answer 1

If we can find [APQ] / ADE], then we can get the required ratio as

[PDFQ][APQ]=[ADE]−[APQ][APQ]=([ADE]/[APQ])−11

Now draw PM⊥AEandDL⊥AE. Observe

[APQ]=21AQ.PM,[ADE]=21AE.DL

Further, since PM∥DL, we also get PM/DL = AP/AD. Using these we obtain

[ADE][APQ]=ADAP.AEAQ.

We have

QEAQ=[ECQ][ABQ]=[ECQ][ACQ]=[BCQ][ABQ]+[ACQ]=[BCQ][ABQ]+[BCQ][ACQ]=