28 CHAPTER 1 Advanced Euclidean Geometry
- Let 4 ABC be given with sidesa= 11, b= 8,andc= 8. Assume
thatDandEare on side [BC] such that [AD],[AE] trisectBAĈ.
Show thatAD=AE= 6. - You are given the equilateral trian-
gle with sides of unit length, de-
picted to the right. Assume also
that AF = BD = CE = r for
some positiver <1. Compute the
area of the inner equilateral trian-
gle. (Hint: try using similar trian-
gles and Stewart’s theorem to com-
puteAD=BE=CF.)
1.3 Circle Geometry
1.3.1 Inscribed angles
Lemma. If a triangle 4 ABC is inscribed in a circle with[AB]being a
diameter, thenACB̂ is a right angle.
Proof. The diagram to the right
makes this obvious; from 2θ+ 2φ=
180, we getθ+φ= 90◦.
Inscribed Angle Theorem.
The measure of an angle inscribed
in a circle is one-half that of the
inscribed arc.