Section 5.6 Segments Divided Proportionately To divide something proportionally, means to divide something according to some ratio. Ex 1. p. 260 Note the Property: if a = c, then a + c = a = c b d b + d b d Ex. 2 p. 260 Theorem 5.6.1: If a line is parallel to one side of a triangle and intersects the other two sides, then it divides these sides proportionately. Proof p. 258 9/22/2018 Section 5.6 Nack

Theorems and Corollaries Corollary 5.6.2: When three (or more) parallel lines are cut by a pair of transversals, the transversals are divided proportionally by the parallel lines. See diagram p. 262, Ex. 3, p. 264 Ex. 5,6 Theorem 5.6.3: (The Angle Bisector Theorem): If a ray bisects one angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the two sides that form the bisected angle. Proof p. 263. Ex. 4,5,6 9/22/2018 Section 5.6 Nack

Ceva’s Theorem Theorem 5.6.4: (Ceva’s Theorem) Let D be any point in the interior of ΔABC and let BE, AF, and CG be the line segments determined by D and vertices of ΔABC. Then the product of the ratios of the lengths of the segments of each of the three sides (taken in order from a given vertex of the triangle ) equals 1; that is: AG  BF  CE = 1 GB FC EA Ex. 7 p. 265 9/22/2018 Section 5.6 Nack