21. Construct a Triangle with Equal Base Angles That Are Each Double the Third Angle
21. Construct a Triangle with Equal Base Angles That Are Each Double the Third Angle
This Demonstration constructs an isosceles triangle with base angles that are double the third angle [2].
Construction
1. Draw the line segment .
AB
2. Draw the circle with center and radius .
σ
A
AB
3. Draw the perpendicular to at with as one point of intersection with .
ρ
AB
A
C
σ
4. Let be the midpoint of .
M
AC
5. Let be on the other side of from such that .
E
A
D
DE=DB
6. Let be on such that .
F
AB
AF=AE
7. Let be on such that .
G
σ
GB=AF
8. Then meets the required conditions.
ΔAGB
Using Euclid II.11 [3], is constructed on such that . The point is constructed such that . Since =BFBA, is tangent to the circumcircle of (Euclid, III, 37 [4]). Hence, . The exterior angle theorem applied to gives . So . Hence, is isosceles and . So, . Then . Therefore, , .
F
AB
ABBF==
2
AF
2
BG
G
BG=AF
2
BG
BG
△AFG
∠BAG=∠FGB
△GAF
∠BFG=∠BAG+∠FGA
∠BFG=∠BGA
△BGF
GF=BG=AF
∠FGA=∠FAG
∠BGA=2∠BAG
∠BAG=
°
36
∠AGB=
°
72