ࡱ> %'$@ 4bjbjFF "6,,4.@.......$Rt9.......TDDD...D.DD:E,.4 TeB:q j0{R(..D..... Db Triangles, Part 1 Things you should know: Triangle Inequality a+b>c *Extended Law of Sines*  EMBED Equation.3  Law of Cosines a2=b2+c2-2bc cosA Pythagorean Theorem a2+b2=(>,<) c2 (acute, obtuse) Area formulas  EMBED Equation.3  Special intersections within triangles Angle bisectors- incenter, perpendicular bisectors- circumcenter, medians- centroid, altitudes- orthocenter Angle bisectors divide the opposite side in the proportion of the other 2 sides Similarity and Congruency *Cevian- a segment from a vertex to a point on the opposite side Notation- [ABC]=area of triangle ABC. Problems: (Difficulty * easy ***** hard) Draw a picture! A=bh, and proportional areas of triangles: *If the bases are the same the ratio of the areas is the ratio of the __________. If the heights are the same the ratio of the areas is the ratio of the __________. In triangle ABC, if D is on segment BC, and E is on segment AD, A=BCADsin(_______) Let  EMBED Equation.3  Evaluate the following:  EMBED Equation.3  **In triangle ABC, D, E, and F are on BC, CA, and AB such that CD, AE, and BF are one third of their respective sides. AD and CF intersect in N1, AD and BE intersect in N2 and BE and FC intersect at N3. AN2:N2N1:N1D=3:3:1. and similarly for lines BE and CF. [N1N2N3]/[ABC]=? **[AMY 2006] Let ABC be an equilateral triangle, and D, E points on sides AB and AC, respectively such that AD=CE. Let segments BE and CD meet at F. If [ABC]=7 and [BCF]=2, find BD/DA. ***Points D, E, F lie on sides BC, CA, AB of triangle ABC such that  EMBED Equation.3 . Let KLM be the triangle enclosed by AD, BE, CF. Express in terms of k the ratio [KLM]/[ABC]. ***In triangle ABC, D is the midpoint of side BC. Point F lies on side AB. Segments AD and CF meet at E. Compute  EMBED Equation.3 . **Three cevians AD, BE, and CF of triangle ABC are concurrent at P. Compute  EMBED Equation.3 . ***[AIME 1984] A point P is chosen in the interior of triangle ABC so that when lines are drawn through P parallel to the sides of triangle ABC, the resulting smaller triangles have areas 4, 9, 49. Find [ABC]. ****[AIME 1989] Three cevians AD, BE, and CF of ABC are concurrent at P. Given AP=6, BP=9, PE=3, CF=20, find [ABC]. hint: First find PF. Then draw some segments and look for similar triangles. What can you conclude (about D)? The tricky part is letting Q be P reflected over D. What can you say about APCQ given its diagonals? 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