Chapter 4: Theorems of Ceva and Menelaus

Question 1

What are the properties of directed distances?

Answer 1

-Given pts A, b on a directed line l”
AB_line = {+AB if A is before B on l
{0 if A=B
{-AB if B is before A on l
-1. AB_line = - BA_line
2. AB_line + BC_line = AC_line
3. If AB_line = AC_line, then C=B

Question 2

What are the properties of directed ratios?

Answer 2

-For A,B,C collinear
1. If C is between A and B, then AC_line / CB_line = +AC/CB
2. If C is external to AB, then AC)line / CB_line = -AC/CB
-For C,D on another line, CD||AB
AB_line / CD_line = {+AB/CD, ABCD is non-s. quad.
{-AB/CD, ABCD is simple

Question 3

Define Cevian of the triangle.

Answer 3

Given a triangle, any line through a vertex and a non-vertex pt of the opposite side (or extended side) is called a cevian of the triangle.
Internal cevian and external.

Question 4

Define a Menelaus pt.

Answer 4

A non-vertex pt of a side or extended side is called a Menelaus pt.

Question 5

Define a Transversal to the triangle.

Answer 5

A line passing through each of the (extended) sides of a triangle, but none of the vertices.
Internal transversal and external.

Question 6

Thm 4.3.2:

Answer 6

Two cevians that pass through the interior of a triangle are not parallel.
Pf provided.

Question 7

Thm 4.2.1 (Ceva’s Thm):

Answer 7

Let AD, BE and CF be lines from vertices A,B,C of a triangle to non-vertex pts D,E,F on the opposite sides,
Then, these cevians are either concurrent or parallel iff
AF_line / FB_line x BD_line / DC_line x CE_line / EA_line = +1.
Pf provided (long, but easy).

Question 8

New pfs for old concepts…

Answer 8

Ex 4.3.4: Internal bisector of a triangle are concurrent.
Ex 4.3.3: The medians of a triangle are concurrent.
Ex 4.3.5 (txt): The altitudes of a triangle are concurrent.
Ex: The medians of a triangle trisect each other.

Question 9

Thm 4.2.2 (Menelaus’ Thm):

Answer 9

Let D,E,F be Menelaus pts on (extended) sides BC, CA, AB of triangle ABC, resp. The pts D, E, F are collinear iff
AF_line / FB_line x BD_line / DC_line x CE_line / EA_line = -1.
Pf provided.

Question 10

Ex 4.4.1:

Answer 10

The internal angle bisectors of two angles in a triangle and the external angle bisector of the third meet the opposite (extended) sides in 3 collinear pts.
Soln provided.

Question 11

Ex 4.4.2 (Simpson’s Thm):

Answer 11

For P on the circumcircle, the perpendiculars from P to BC, AC, AB meet these (extended) sides at 3 collinear pts.
Soln provided.