Exercise on a special cyclic quadrilateral

Suppose that ABCD is a cyclic quadrilateral where triangle ABC is equilateral.

• What is angle CDA?
• What is angle CDB?
• What is angle BDA?
• If a point P is such that angle CPA = 120 degrees, where are the possible locations of P?
  

More Special Properties of the Napoleon Figure

In this figure, the triangles with centers X, Y, Z are equilateral.

• Draw the circumcircles of each of the equilateral triangles.  What do you see?
• Draw the 3 segments AA', BB', CC'.  What do you see?  Check lengths and angles.

Proofs:

• Prove AA' is congruent to BB' by finding a rotation that takes one segment to the other.  This also will establish the angle between them.
• Let Q be the point of intersection of these two segments.  Can you prove that Q is on one or more of the circumcircles?
• Let P be the point of intersection of two of the circumcircles.  Can you find relationships that show that P is on the other circumcircle?