Plane geometry: constructing techniques (1)

"The journey of a thousand miles begins with a single step" (Lao Tzu)

The constructing techniques (drawing) by basic rulers and compasses is fairly simple. Only when you have mastered the use of this technique, you can then construct more complex geometric figures accurately, scientifically and clearly.

(1) Constructing a perpendicular bisector of a given line segment AB

Perpendicular bisector of the line segment is a line perpendicular to the given line segment at its midpoint . To draw the perpendicular bisector of the line segment AB, we just draw two circles of centre A and B respectively and of equal radius R ( R > AB / 2 ) . Two circles intersect at two points I and K. The line passing through these 2 points I, K is the perpendicular bisector of AB .

(2) Constructing a line through point A and perpendicular to a given line (d)

To draw the line through point A and perpendicular to line (d), draw a circle of center A. This circle cuts the straight line (d) at two points B and C. Draw two circles of centre B and C with radius R = AB . Two circles intersect at points A and D. The straight line AD is the line we need to construct ( AD is perpendicular to (d) ) .

(3) Constructing bisector of a given angle A

Method: Draw a circle of center A and of any radius R ( R1 > 0 ). This circle cuts the sides Ax, Ay of the given angle at two points I and J. Draw two circles of center I and J, with radius R2 such as the two circles can intersect. Let K be the intersection of two circles of center I, J as above. The straight line passing through the two points A and K is the bisector of angle A.

(4) Constructing a tangent to the circle of center O from a given external point A

Draw a circle of diameter AO which cuts the circle of center point O at two points P and Q. The two straight lines AP and AQ are two tangents to the circle from point A.

(5) Divide a line equally in three

PROBLEM: From point O outside the line segment AB , construct two straight lines OM , ON ( M , N are the two points on line segment AB ) that divide AB into three equal line segments .

Solution: Draw a circle of center O and of radius AO that cuts the straight line AO at P (different from point A). Then, Draw a circle of center P and of radius AO that cuts the straight line AO at another point Q (Q is different from P). Draw the straight line QB. Through O , P, draw the straight lines that are parallel to QB , cutting AB at M , N respectively. The straight lines OM, ON are the ones we need to construct.

27.1.2014

Khuc Trung Kien

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