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Basic Constructions and Points of Concurrency - PowerPoint PPT Presentation

Basic Constructions and Points of Concurrency. Objectives. What is construction? Who invented this tool commonly used in geometry? Circumcenter Incenter Centroid Orthocenter Euler Line. TOOLS NEEDED. COMPASS STRAIGHT EDGE PENCIL PAPER YOUR BRAIN (THE MOST IMPORTANT TOOL).

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Basic Constructions and Points of Concurrency

• What is construction?

• Who invented this tool commonly used in geometry?

• Circumcenter

• Incenter

• Centroid

• Orthocenter

• Euler Line

• COMPASS

• STRAIGHT EDGE

• PENCIL

• PAPER

• YOUR BRAIN (THE MOST IMPORTANT TOOL)

• the drawing of geometric items such as lines and circles using only a compass and straightedge. Very importantly, you are not allowed to measure angles with a protractor, or measure lengths with a ruler.

• The compass is a drawing instrument used for drawing circles and arcs. It has two legs, one with a point and the other with a pencil or lead. You can adjust the distance between the point and the pencil and that setting will remain until you change it.

• Note: This kind of compass has nothing to do with the kind used find the North direction when you are lost.

Straightedge

• A straightedge is simply a guide for the pencil when drawing straight lines. In most cases you will use a ruler for this, since it is the most likely to be available, but you must not use the markings on the ruler during constructions. If possible, turn the ruler over so you cannot see them.

Euclid, the ancient Greek mathematician is the acknowledged inventor of geometry.

He did this over 2000 years ago, and his book "Elements" is still regarded as the ultimate geometry reference.

Father of Geometry

• Why didn't Euclid just measure things with a ruler and calculate lengths so as to bisect them?

• The Greeks could not do arithmetic. They had only whole numbers, no zero, and no negative numbers like the Roman numerals. In short, they could perform very little useful arithmetic.

• So, faced with the problem of finding the midpoint of a line, they could not do the obvious - measure it and divide by two. They had to have other ways, and this lead to the constructions using compass and straightedge. It is also the reason why the straightedge has no markings. It is definitely not a graduated ruler, but simply a pencil guide for making straight lines. Euclid and the Greeks solved problems graphically, by drawing shapes, as a substitute for using arithmetic.

When two lines intersect at one point, we say that the lines

are intersecting. The point at which they intersect is

the point of intersection.

(nothing new right?)

Well, if three or more lines intersect at a common point, we say that the lines are concurrent lines.The point at which these lines intersect is called the point of concurrency.

• Concurrent Lines – Three or more lines that intersect at a common point.

• Point of Concurrency – The point where concurrent lines intersect.

Both sides are congruent- make sure you see this or it is NOT a perpendicular bisector

Perpendicular Bisector

midpoint and perpendicular

Angle Bisector

cuts the angle into 2 equal parts

Both sides are congruent

Median

vertex to midpoint

Altitude

vertex to opposite side and perpendicular

A

A

A

A

A

B

B

B

B

B

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MedianAltitudeNone Angle Perp

Bisector Bisector

A perpendicular bisector cuts a line exactly in half at a right angle.

The point where the 3 perpendicular bisectors meet is called the circumcentre.

The circle which passes through the vertices of the triangle with the circumcentre as its centre, is called the circumcircle.

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The bisectors of the angles of a triangle are concurrent and the point of intersection is the centre of an inscribed circle.

An angle bisector is a line that cuts an angle exactly in half.

The point where the 3 medians meet is called the centroid.

A median of a triangle is a line from a vertex to the mid-point of the opposite side.

The point where the 3 altitudes meet is called the orthocenter.

An altitude of a triangle is a line from a vertex perpendicular to the opposite side.

The orthocenter, circumcenter, and the centroid are COLLINEAR in EVERY triangle!

Concurrent Lines

3 or more lines that intersect at a common point

Point of Concurrency

The point of intersection when 3 or more lines intersect.

Type of Line SegmentsPoint of Concurrency

Perpendicular Bisectors Circumcenter

Angle Bisectors Incenter

Median Centroid

Altitude Orthocenter

• Facts to remember:

• The circumcenter of a triangle is equidistant from the

• vertices of the triangle.

• Any point on the angle bisector is equidistant from the sides of the angle (Converse of #3)

• Any point equidistant from the sides of an angle lies on

• the angle bisector. (Converse of #2)

• The incenter of a triangle is equidistant from each side of the triangle.

• The distance from a vertex of a triangle to the centroid is 2/3 of the median’s entire length. The length from the centroid to the midpoint is 1/3 of the length of the median.