Math 487 Lab 2: Distance from Lines and Tangent Circles

Outline of Lab

Part 1. What is a Voronoi diagram or a Dirichlet domain?

Part 2. How to construct circles tangent to two lines. What is the connection between tangent circles and the concept of the distance from a point to a line?

Part 3. Concurrence of angle bisectors and construction of inscribed and excribed circles.

Part 1.

• Read and work through the Dirichlet section of this pdf document from Geometry Through the Circle.

• Construct the figure with rays for 3 points A, B, C. In this case the figure will probably not stay correct if you drag the points too far. Do you see why?

• Download and explore this Dirichlet gsp file after you have done some of your own constructions.

Part 2. Distance from a Point to a Line and tangent circle

1. Work through the investigations in this pdf file from Geometry Through the Circle.

What you should take away from this is a set of answers to these questions:

1. In a right triangle, why is the hypotenuse the longest side?

2. Why is the shortest distance from a point C to the points on a line AB given by the distance from C to the foot E of the perpendicular from C to line AB?

3. Explain why the figure with the tangent circle in this investigation illustrates visually the point above. Why does the line AB intersect the circle only at one point E?

4. If a circle with center O intersects the line at two points U and V, why does the perpendicular bisector of UV pass through the circle of the circle? What is the distance from the center to of the circle to the line measured by the distance from O to the midpoint of UV?

Part 3.

A. Inscribed Circles

• Draw a triangle. Construct the circle that is inside the triangle and tangent to all 3 sides. This is called the inscribed circle of the triangle, or the incircle.

B. Angle Bisectors of a triangle (interior and exterior angles)

• In a new sketch, draw 3 points A, B, C and construct the 3 lines a, b, c (NOT SEGMENTS) connecting each pair of these points. This is an (extended) triangle.
• The lines that bisect the interior and exterior angles of the triangle.
• Observe that at any point of intersection of two of these bisectors, there is a third bisector through the same point. Why is this true? How many such points of intersection are there? Why?

C. Excircle and Incircles

• Construct all circles that are tangent to each of the 3 lines. How many are there?

Reference: B&B, pp. 182-3 and 250-1.