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Texas A&M University Department of Mathematics Volodymyr Nekrashevych Spring 2011 MATH 367 Homework 4 Issued: 03.25 Due: 04.07 4.1. Suppose that in 4ABC we have AB ∼ = BC. Let A1 and C1 be the midpoints of the segments CB and AB, respectively. Prove that AA1 ∼ = CC1 . 4.2. Let D be an interior point of an angle ∠ABC. Prove that perpendiculars from D −→ −−→ to the sides BA and BC of ∠ABC are congruent if and only if D belongs to the bisector of ∠ABC. 4.3. Let A1 be the midpoint of the side BC of 4ABC. Prove that length of AA1 is less −−→ than 12 (AB + AC). (Hint: consider the point A2 on AA1 such that AA1 = A1 A2 and A − A1 − A2 , and consider 4AA2 B.) 4.4. Let A, B, C, D be four points such that ∠ABC ∼ = ∠BCD, and suppose that A and ←→ ←→ ←→ D are on opposite sides of the line BC. Prove that lines AB and CD do not intersect.