Geometry with Trigonometry

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|◦.