MathDB

4

Part of 2011 AIME Problems

Problems(2)

Triangle and angle bisectors

Source:

3/18/2011
In triangle ABCABC, AB=125,AC=117AB=125,AC=117, and BC=120BC=120. The angle bisector of angle AA intersects BC‾\overline{BC} at point LL, and the angle bisector of angle BB intersects AC‾\overline{AC} at point KK. Let MM and NN be the feet of the perpendiculars from CC to BK‾\overline{BK} and AL‾\overline{AL}, respectively. Find MNMN.
geometrygeometric transformationhomothetytrigonometryangle bisectortrig identitiesLaw of Sines
Angle bisectors [2011.II.4]

Source:

3/31/2011
In triangle ABCABC, AB=2011ACAB=\frac{20}{11} AC. The angle bisector of ∠A\angle A intersects BCBC at point DD, and point MM is the midpoint of ADAD. Let PP be the point of the intersection of ACAC and BMBM. The ratio of CPCP to PAPA can be expressed in the form mn\dfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
ratiogeometryanalytic geometrygraphing linesslopeAMCAIME