Advanced High-School Mathematics

30 CHAPTER 1 Advanced Euclidean Geometry

Exercises

1. In the diagram to the right, the arc
   AB ̆has a measure of 110◦ and the
   measure of the angle ACB̂ is 70◦.
   Compute the measure ofADB̂.^6


2. Let [AB] be a diameter of the circle C and assume that C is a
   given point. IfACB̂ is a right angle, thenC is on the circleC.


3. Let C be a circle having center
   O and diameter d, and let A, B,
   and C be points on the circle. If
   we set α = BAĈ , then sinα =
   BC/d. (Hint: note that by the
   inscribed angle theorem, BAĈ =
   POĈ. What is the sine ofPOĈ ?)


4. In the given figure AF = FC and
   PE=EC.
   (a) Prove that triangle 4 FPA is
   isosceles.
   (b) Prove thatAB+BE=EC.


5. A circle is given with centerO. The
   pointsE, O, B, D, and E are col-
   inear, as areX, A, F, andC. The
   lines (XC) and (FD) are tangent
   to the circle at the pointsAandD
   respectively. Show that
   (a) (AD) bisectsBAĈ ;
   (b) (AE) bisectsBAX̂.