Flashcards in Chapter 2: Concurrency Deck (42):

1

## Define Parallelogram.

### A parallelogram is a quad. whose opposite sides are parallel. A parallelogram whose sides are equal is called a rhombus.

2

## Thm 2.3.1:

###
In a parallelogram:

1. Opposite sides are congruent

2. Opposite angles are "

3. The diagonals bisect each other.

Pf provided.

3

## Thm 2.3.3/5:

###
A simple quad. is a parallelogram if any one of the following hold:

1. Opposite sides are congruent

2. Opposite angles are "

3. The diagonals bisect each other.

4. One pair of opposite sides are congruent and parallel.

Pf provided.

4

## Thm 2.4.2 (Midline Thm):

###
If P and Q are the respective midpoints of AB and AC, then PQ is parallel to BC and PQ=BC/2.

Pf provided.

5

## Thm 2.4.5:

###
Let P be the midpt of AB and Q a pt on AC s.t. PQ||BC. Then Q is the midpt of AC.

Pf provided.

6

## Define concurrent.

### A family of lines is called concurrent at a pt P if they all pass through P.

7

## Define perpendicular.

### Two lines that intersect at right angles (⊥).

8

## Thm 2.1.1/Corollary 2.1.2:

###
Assume l_1 ⊥ m_1, and l_2 ⊥ m_2. Then l_1 || l_2 iff m_1 || m_2.

Pf provided.

9

## Define the perpendicular bisector.

###
The perpendicular (right) bisector of AB is the line m passing through the midpt M of AB.

Fact: a pt X ϵ m iff XA=XB.

Pf provided.

10

## Thm 2.1.3:

###
The perp. bisector of the sides of a triangle are concurrent.

Pf provided.

11

## We call the pt O the ___ of △ABC, and the circle C(O,r) with centre O and radius r=OB=PA=PC the ___.

### Circumcentre and circumcircle.

12

## Define chord.

### A chord of a circle is any line segment joining two pts on it.

13

## Define line tangent.

### A line is tangent to a circle if it intersects it exactly once.

14

## Define distance.

###
The distance between 2 objects is the shortest distance between points on those objects. Denoted dist(X,Y).

Fact: for a pt P that is not on line l, there is a unique pt Q ϵ l s.t. PQ ⟂ l. PQ = dist(P,l).

15

## Thm 2.2.1:

###
For a circle:

1. The perp. bisector of any chord of this circle passes through O.

2. If a given chord is not a diameter, the line joining O to the midpt of the chord id the perp. bisector of the chord.

3. The unique line from O that is perpendicular to a chord is its perp. bisector.

4. A line is tangent to circle at a point P iff it is perp. to radius OP.

Pf provided.

16

## Define angle bisector.

### Given a no-reflex angle ∠ABC, a ray BD s.t. ∠ABD=∠CBD is called and angle bisector of ∠ABC.

17

## Thm 1.5.6 (Characterisation of the angle bisector):

###
Given a non-reflex angle ∠ABC, a pt X is on the angle bisector of ∠ABC iff dist(X, AB) = dist(X,CB).

Pf provided.

18

## Thm 2.2.2:

###
The internal bisectors of the angles of a triangle are concurrent.

Pf provided.

19

## We call the pt I (common pt of intersection which gives the centre of a circle that can be inscribed in △ABC) the ___ and the circle its ___.

### Incentre and incircle.

20

## Prop 2.2.3:

###
Each pair of external angle bisectors of a triangle intersect.

Pf provided.

21

## Thm 2.2.4:

###
The external bisectors of two external angles of a triangle and the internal bisector of the third angle are concurrent.

Pf provided.

22

## What are the pt (at which the external bisectors of two external angles and the internal bisector of the third angle are concurrent) and the circle called?

### Excenter and excircle.

23

## Define altitude.

### It is a line passing through a vertex of a triangle which is perp. to the opposite side.

24

## Thm 2.3.6:

###
The altitudes of a triangle are concurrent.

Pf provided.

25

## What is the pt of concurrency of the altitudes called?

### The orthocentre (it can be inside, outside, or on a vertex of a triangle and is often denoted H).

26

## Define median.

### A line that passes through a vertex and the midpt of the opposite side of a triangle.

27

## Lemma (medians):

###
Any two medians of a triangle trisect each other at their pt of intersection.

Pf provided.

28

## Thm (medians):

###
The medians of a triangle are concurrent.

Pf provided.

29

## What is the pt called where 3 medians meet?

### Centroid (Physics centre of mass).

30

## In an isosceles triangle with AB=AC, what is special about the right bisector of BC, the angle bisector of ∠A, the altitude from A and the median through A?

### They all coincide.

31

## Provide notation for Thales' Thm (1.3.2).

###
For a circle C(O,r) and A, B, C on it, ∠ABC is an inscribed angle

-The arc from A to C NOT passing through B is denoted AC with arc over

-∠AOC is called the central angle of AC_arc

-We denote m(AC_arc) the measure of this angle (can be more than 180)

32

## Thales' Thm (1.3.6):

###
Given distinct A, B, C on C(O,r), ∠ABC = m(AC_arc) /2

Pf provided.

33

## Corollary 1.3.7:

###
In a given circle:

1. All angles inscribed on the same arc are equal

2. All inscribed angles from congruent arcs are equal.

3. An angle inscribed in a semi-circle is a right-angle.

34

## If A,B,C,D on C(O,r) and AB=CD, then...?

### AB_arc = CD_arc and m(AB_arc) = m(CD_arc)

35

## Thm:

###
For △ABC, with D the midpt of AC: ∠ABC = 90 iff AD=BD=DC.

Pf provided.

36

## Tangent Chord Thm:

###
Then angle between a tangent and a chord is half the measure of the enclosed arc.

Pf provided.

37

## What are some more corollaries? Pfs provided.

###
1. ∠APD = (m(AD) + m(BC)) /2 for pt inside

2. ∠APD = (m(AD) - m(BC)) /2 for pt outside

3. ∠APC = (m(AC) - m(BC)) /2 for pt outside and one line tangent

4. ∠APC = (m(ABC) - m(ADC)) /2 for pt outside and two lines tangent

38

## Define Cyclic polygon.

### A polygon which can be inscribed in a circle (that is, each of its vertices is on the circle). The circle is its circumcircle and its center the circumcenter.

39

## Thm (Simple Cyclic Quadrilaterals):

###
A simple quadrilateral is cyclic iff a pair of opposite angles sum to 180 degrees.

Pf provided.

40

## Example 1.3.13 (Cyclic Non-simple Quads/Bowtie Thm):

###
A non-simple quad. is cyclic iff a pair of opposite angles are equal.

Pf provided.

41

## Refer to Construction videos for methods to properly construct angles, circles, etc... and explanation of Thales' Locus (Thm 2.5.4).

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