Flashcards in Chapter 4: Theorems of Ceva and Menelaus Deck (11):

1

## What are the properties of directed distances?

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-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

2

## What are the properties of directed ratios?

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-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

3

## Define Cevian of the triangle.

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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.

4

## Define a Menelaus pt.

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

5

## Define a Transversal to the triangle.

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

Internal transversal and external.

6

## Thm 4.3.2:

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

Pf provided.

7

## Thm 4.2.1 (Ceva's Thm):

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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).

8

## New pfs for old concepts...

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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.

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## Thm 4.2.2 (Menelaus' Thm):

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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.

10

## Ex 4.4.1:

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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.

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