Two circles intersect at points **C** and **D**. Point **A** lies on Circle 1, point **B** lies Circle 2. Find the positions of **A** and **B** (on their given circle) such that segment **AB** is at its maximum possible length, and points **A**, **C**, and **B** are collinear.

* Circle 1 Circle 2*

**Answer**

We first connect points **A**, **B**, and **D** to construct triangle **ABD**. What happens with the angles of the triangle as **A** and **B** change their position?

* Circle 1 Circle 2*

Angle **BAD** is the same as **CAD**, and angle **CAD** is half the measure of arc (**CD**) (*Interior Angle Theorem*). Therefore as we move point **A** along its circle, the measure of angle **BAD** does not change.

Same argument can be made for angle **ABD**, and we can conclude that the size of the angles of triangle **ABD** does not change as **A** and **B** slide along the two circumferences.

* Circle 1 Circle 2*

Any two versions of **ABD** are similar triangles, and the version with the longest possible length of **AB** has the longest possible lengths for **AD** and **BD** too.

When is **AD** the longest? When **A** and **D** are the end points of a diameter of *Circle 1*.

When is **BD** the longest? When **B** and **D** are the end points of a diameter of *Circle 2*.

What is the measure of the angle determined by **AB** and **CD**? Both angle **ACD** and angle **BCD** are inscribed in a semicircle, so they measure 90°.