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.