Chapter 2: Concurrency Flashcards Preview

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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).

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

42

Refer to Power of a Point videos for Thm 3.6.1, 3.6.2, 3.6.4 and examples.

Refer to Power of a Point videos for Thm 3.6.1, 3.6.2, 3.6.4 and examples.