Chapter 3

GEOMETRICAL CONSTRUCTIONS

Dr. Pham Minh Tuan

Parallel & Perpendicular lines

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 Midpoint, Bisector

Midpoint:

Bisector:

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 Divide a line into any equal parts
The process to divide a line into n equal partitions:
Draw a line Ax through A
Divide Ax into n equal partitions, number the marks as 1', 2', …, n'
Join the n' mark on Ax to B
Draw lines parallel to n'B from the marks on Ax to intersect AB at
corresponding points 1, 2, …, (n-1)

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 Regular polygons inscribed in circles
Equilateral triangle:

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 Regular polygons inscribed in circles
Quadrilateral (Square):

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 Regular polygons inscribed in circles
Regular hexagon:

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 Divide a circle into equal parts
Divide a circle into 3, 6, … equal parts:

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 Divide a circle into equal parts
Divide a circle into 2, 4, 8 … equal parts:

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 Divide a circle into equal parts
Divide a circle into 5, 10 equal parts:
Draw two normal diameter, i.e, AB and CD.
Determine the midpoint M of the radius OA.
The arc (M, MC) intersects OB at N.
CN is the length of the regular pentagon’s edge.
ON is the length of the regular decagon’s edge.

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 Divide a circle into equal parts
Divide a circle into 7 (9,11,13,…) equal parts:
Draw two normal diameter, i.e, AB and CD.
Draw the arc (D, CD) to cut AB at E và F.
Divide CD into 7 (or 9, 11, 13) equal partitions, number the marks.
Join E, F with even marks (2, 4, 6) or odd marks (1, 3, 7). These
lines intersect the circle at vertices of the regular heptagon.

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 Slope
The slope of a line is the ratio of the amount that y increases as x
increases some amount. Slope tells you how steep a line is, or how
much y increases as x increases. The slope is constant (the same)
anywhere on the line.
The slope of two lines AB and AC is measured by the tangent of
the angle α between them.
S = BC/AC = tang(α)

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 Taper
Definition of taper:
K = (D – d)/h = 2tang(α)

A cone shape with a taper of K is an isosceles trapezoid with

dimensions of two parallel edges being D and d, two lateral edges
have a slope of K/2 compared to the altitude of the trapezoid.
Some standard values of taper: 1:3, 1:5, 1:7, 1:8, 1:10, 1:12, 1:15,
1:20, 1:30, 1:50, 1:100, 1:200.
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 Lines tangential to a given circle
General method:
 Join C to O
 Determine the midpoint I of OC
 Draw the circle (I, OC/2) to
intersect the given circle at T1
and T2
 CT1 and CT2 are 2 tangent lines
of the given circle

Using triangle ruler:

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 Lines tangential to given circles/arcs
External tangents:
 Construct the circle (O1, R1 – R2)
 Draw the line O2A tangential to the circle (O1, R1 – R2)
 Draw the line O1A to intersect the circle (O1, R1) at T1
 Draw the line O2T2 parallel to the line O1T1
 T1T2 is the tangent line of both circles (O1, R1) and (O2, R2)
 Apply similar steps to construct the other tangent line T1'T2'

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 Lines tangential to given circles/arcs
Internal tangents:
 Draw the circle (O2, R1 + R2), other steps are similar to the case of
external tangents.
 If O1O2 > R1 + R2: there are 2 internal tangent lines.
 If O1O2 = R1 + R2: there is 1 internal tangent line at the contact point.
 If O1O2 < R1 + R2: there is no internal tangent line.

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 An arc tangential to given lines
 Two lines d1 and d2 parallel to each other:
 Draw a line perpendicular to d1, d2 and to intersect them at T1, T2
 Define the midpoint O of the line T1T2
 Draw the circular arc T1T2 with the center O and the radius of OT1 or OT2

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 An arc tangential to given lines
 Two lines d1 and d2 intersect each other:
 From inside the area created by d1 and d2, draw d'1//d1 and d'2//d2, offset
from d1 and d2 a distance of R (radius of the tangent arc)
 The intersection O of d'1 and d'2 is the center of the tangent arc
 From O draw lines perpendicular to d1 and d2 at T1 and T2
 Draw the circle (O, R) join T1 and T2

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An arc tangential to given line and
another arc/circle
External tangent:
 Construct the line d' parallel to the given
line d and to offset from d a distance of R
(radius of the tangent arc)
 Draw the circle (O1, R + R1) to intersect
d' at O (center of the tangent arc)
 OO1 intersects (O1, R1) at T1. Draw OT2
perpendicular to d and intersect d at T2
 Draw the arc (O, R) to join T1 and T2

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An arc tangential to given line and
another arc/circle
Internal tangent:
 Construct the line d' parallel to
the given line d and to offset from
d a distance of R (radius of the
tangent arc)
 Draw the circle (O1, R − R1) to
intersect d' at O (center of the
tangent arc)
 OO1 intersects (O1, R1) at T1.
Draw OT2 perpendicular to d and
intersects d at T2
 Draw the arc (O, R) to join T1
and T2

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 An arc tangential to two given arcs
External tangents:
 Construct 2 arcs (O1, R + R1) and (O2, R + R2) where R is the radius of the
tangent arc, these 2 arcs intersect each other at O (center of the tangent arc)
 OO1 intersects (O1, R1) at T1, OO2 intersects (O2, R2) at T2
 Draw the arc (O, R) to join T1 and T2

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 An arc tangential to two given arcs
Internal tangents:
 Construct 2 arcs (O1, R − R1) and (O2, R − R2) where R is the radius of the
tangent arc, these 2 arcs intersect each other at O (center of the tangent arc)
 OO1 intersects (O1, R1) at T1, OO2 intersects (O2, R2) at T2
 Draw the arc (O, R) to join T1 and T2

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 An arc tangential to two given arcs
External and internal tangents:
 Depending on the given case of external and internal tangents, construct 2
arcs (O1, R − R1) and (O2, R + R2) where R is the radius of the tangent arc,
these 2 intersect each other at O (center of the tangent arc)
 OO1 intersects (O1, R1) at T1, OO2 intersects (O2, R2) at T2
 Draw the circle (O, R) to joint T1 and T2

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END OF CHAPTER 3