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The circumscribed circle

Create a triangle.

Given any triangle, it is always possible to find a circle such that all the vertices of the triangle lie on the circle, the so called circumscribed circle.

Start by finding a circle whose midpoint is equidistant from two of the points. On what line must the midpoint of the circle lie?

The inscribed circle

Given any triangle, it is always possible to find a circle inside the triangle such that the circle is tangent to each of the three sides of the triangle, the so called inscribed circle.

Start by considering a circle that is tangent to two of the sides. On what line must the midpoint of the circle lie? Show hint!

Three circles

Given three points. Construct three circles tangent to each other.

A square inscribed in a triangle

Given an acute triangle you can always find a square having its vertices on the sides of the triangle.

Solve the problem in following steps:

Is there an easy way of inserting a square in a triangle such that two of the vertices lie on sides of the triangle?

Is there an easy way of inserting a square in a triangle such that three of the vertices lie on sides of the triangle?

In what way does the fourth vertex move when the square is altered? Find the pattern! Make a conjecture about the position of the fourth vertex.

Napoleon's Theorem