Assignment 3 (due Friday, 1/28)

3.1 Dividing a segment by ratio k

a) Given points A and B and a constant k > 0, use algebra to prove that there are exactly two points C and D on line AB so that |AC/BC| = k = |AD/BD| with an exception for one value of k.

b) What is the exceptional value and where is the point C for which |AC/BC| = k in this case

c) In the general case, show that one of the points, C, is on the segment AB and AC/BC = -k.

d) Show the other point D is on the line exterior to the segment AB and AD/BD = +k.

3.2 Division of a segment by inversion

Let B be the inversion of point A in circle c. Let O be the center of c and r be the radius. Also, let the line OA intersect c in points C and D.

· Prove that AC/AD = BC/DB.

Note that these are signed ratios.

Definition: If this equation is true, A and B are said to divide CD harmonically.

Hint: You can use coordinates on the line if you like, although the computation is quite possible without coordinates.

3.3 Some algebraic rearrangement

If points A, B, C, D on a line satisfy AC/AD = BC/DB, then show this is also true:

• CA/CB = DA/BD.
• Restate this fact using the harmonic division terminology.

3.4 Angle Bisector Ratios

Given a triangle PCD, let the interior angle bisector of angle DPC intersect line CD at point A and let the exterior angle bisector intersect line CD at point B.

a) Prove that |AC|/|AD| = |PC|/|PD|.

b) Prove that |BC|/|BD| = |PC|/|PD|.

c) Conclude that |AC|/|AD| = |BC|/|BD|. Restate this fact using harmonic division terminology.

Note: There is one special case.

Hint: If |PC|<|PD| as in the figure, reflect C across PB, look for parallels and transversals.