# 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