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MATH 241 > Chapter 2: Concurrency > Flashcards

Chapter 2: Concurrency Flashcards

1
Q

Define Parallelogram.

A

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

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2
Q

Thm 2.3.1:

A 
In a parallelogram:
1. Opposite sides are congruent
2. Opposite angles are "
3. The diagonals bisect each other.
Pf provided.
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3
Q

Thm 2.3.3/5:

A

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.

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4
Q

Thm 2.4.2 (Midline Thm):

A

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

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5
Q

Thm 2.4.5:

A

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.

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6
Q

Define concurrent.

A

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

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7
Q

Define perpendicular.

A

Two lines that intersect at right angles (⊥).

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8
Q

Thm 2.1.1/Corollary 2.1.2:

A

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

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9
Q

Define the perpendicular bisector.

A 
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.
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10
Q

Thm 2.1.3:

A

The perp. bisector of the sides of a triangle are concurrent.
Pf provided.

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11
Q

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

A

Circumcentre and circumcircle.

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12
Q

Define chord.

A

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

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13
Q

Define line tangent.

A

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

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14
Q

Define distance.

A

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).

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15
Q

Thm 2.2.1:

A

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.

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16
Q

Define angle bisector.

A

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

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17
Q

Thm 1.5.6 (Characterisation of the angle bisector):

A

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
Q

Thm 2.2.2:

A

The internal bisectors of the angles of a triangle are concurrent.
Pf provided.

19
Q

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 ___.

A

Incentre and incircle.

20
Q

Prop 2.2.3:

A

Each pair of external angle bisectors of a triangle intersect.
Pf provided.

21
Q

Thm 2.2.4:

A

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

22
Q

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?

A

Excenter and excircle.

23
Q

Define altitude.

A

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

24
Q

Thm 2.3.6:

A

The altitudes of a triangle are concurrent.

Pf provided.

25
Q

What is the pt of concurrency of the altitudes called?

A

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

26
Q

Define median.

A

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

27
Q

Lemma (medians):

A

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

28
Q

Thm (medians):

A

The medians of a triangle are concurrent.

Pf provided.

29
Q

What is the pt called where 3 medians meet?

A

Centroid (Physics centre of mass).

30
Q

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?

A

They all coincide.

31
Q

Provide notation for Thales’ Thm (1.3.2).

A

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
Q

Thales’ Thm (1.3.6):

A

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

Pf provided.

33
Q

Corollary 1.3.7:

A

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
Q

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

A

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

35
Q

Thm:

A

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

Pf provided.

36
Q

Tangent Chord Thm:

A

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

37
Q

What are some more corollaries? Pfs provided.

A
  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
Q

Define Cyclic polygon.

A

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
Q

Thm (Simple Cyclic Quadrilaterals):

A

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

40
Q

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

A

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

41
Q

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

A

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

42
Q

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

A

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

MATH 241 (5 decks)

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