000RM.dvi

130 Basic geometric constructions

Exercise

1.Given triangleABC, construct the equilateral trianglesBCX,CAY

andABZexternally on the sides of the triangle. JoinAX,BY,

CZ. What can you say about the intersections, lengths, and direc-

tions of these lines (segments)?

2.Show that the 90 ◦angle of a right triangle is bisected by the line

joining it to the center of the square on the hypotenuse.

3.Make a sketch to show that for two given positive quantitiesaand

b,

a+b

2

≥

√

ab≥

2 ab

a+b

.

4.Construct the following diagram.

B A

C D

5.Construct the following diagram.

B A

C D