Sec. 5.6 Area of triangles, and convex quadrilaterals and polygons 79
ThenPQ‖BC.
5.6 LetAB⊥ACand letD=mp(B,C). Prove that|D,A|=|D,B|=|D,C|.
5.7 LetA,B,Cbe non-collinear points and forA∈[B,PandA∈[C,QletPQ‖BC.
Show that then
|A,P|
|A,B|
=
|A,Q|
|A,C|
.
5.8 Suppose thatA,B,Care non-collinear points with|A,B|>|A,C|and letD=
πBC(A). Prove that then
|A,B|^2 −|A,C|^2 =|B,D|^2 −|C,D|^2.
5.9 Suppose thatA,B,Care non-collinear points andDis the mid-point ofBand
C. Prove that then
|A,B|^2 +|A,C|^2 = 2 |B,D|^2 + 2 |A,D|^2.
[Hint. Consider the foot of the perpendicular fromAtoBC.]
5.10 Show that the AAS-principle of congruence in Ex.4.2 can be deduced from
5.2.2 and the ASA-principle.
5.11 Show that the AAS-principle of congruence for right-angled triangles in Ex.4.3
can be deduced from Pythagoras’ theorem and the SSS-principle.
5.12 ForC∈AB, suppose thatmis the line throughCwhich is parallel toAB. Prove
that for any pointD∈ABthe lineADmeetsmin a unique pointE. When,
additionally,D∈IR(|BAC)thenEis on[A,Dand is also onm∩IR(|BAC).
5.13 In a triangle[A,B,C],let|A,B|>|A,C|.LetD∈[A,B be such that|A,D|=
|A,C|. Prove that then
2 |∠BCD|◦=|∠ACB|◦−|∠CBA|◦.