Do all 4 problems.

In construction problems you must WRITE brief descriptions of the major steps of your construction. You are NOT supposed to justify or prove the construction, but you ARE supposed to make clear how the various elements were constructed. So you can refer to long-established constructions very briefly, but use point labels so that we can find your marks. For example "Construct the perpendicular bisector of AB" is plenty detailed, and you can even say, "Construct the circle through ABC using perpendicular bisectors." You do not have to say "Construct the perpendicular bisector of AB by drawing an arc with center A and radius AB and then drawing an arc with center B…."

When stating a definition, use complete sentences and make clear what any letters stand for. Assume your reader knows standard geometry terminology but never heard your definition before. But do not embroider the definition with additional "facts" or theorems beyond the definition, even if they are true. If you can't resist adding something like this, separate it from the definition.

State the definition of inversion of a point in a circle.

State the definition of the power of a point with respect to a circle.

Let c be a circle with center O and radius r. Suppose that A is a point not on c and distinct from O. Let A' be the inversion of A in c. Prove that any circle d through both A and A' is orthogonal to c.

Note: State clearly the definitions and theorems you use in the proof. You do not have to prove the theorems you use, provided that they are not "this theorem" in different words.

In the figure below, construct the circle through A and B which is orthogonal to circle c.

Construct a circle d that is orthogonal to all the three circles c1, c2, c3, whose centers are points O1, O2, and O3.