Geometric Constructions: Triangles, Squares, Polygons, and Tangencies

Posted on May 12, 2024 in Visual arts

Mapping Geometric Shapes

Mapping the Equilateral Triangle Knowing its Side

  1. Draw a line segment AB with the given side value.
  2. Using a compass opening equal to AB, draw two arcs centered at vertices A and B. These arcs will intersect at point C, the vertex opposite side AB.
  3. Connect C with A and B to form the triangle.

Tracing the Square Knowing its Side

  1. Draw side AB with the given value.
  2. Draw a perpendicular line at each of the vertices A and B using a set square and bevel. From A, draw a line forming a 45° angle with side AB. This line will intersect the perpendicular from B at point C.
  3. Draw a line parallel to AB through C to complete the square.

Mapping the Pentagon Knowing its Side

  1. Draw side AB with the given value and bisect it to find point P.
  2. Draw a line perpendicular to AB at B. From B, with radius BA, draw an arc that intersects the perpendicular at point J.
  3. With center P and radius PJ, draw an arc that intersects the extension of AB at point M.
  4. From A, with an opening of AM, draw an arc that intersects the bisector at point D.
  5. Finally, draw arcs with centers D, A, and B, and radius equal to side AB. These arcs will intersect to determine points C and E, the remaining vertices of the pentagon. Connect points C, D, and E with A and B to complete the pentagon.

Tracing the Regular Hexagon Knowing its Side

The regular hexagon is the only regular polygon whose side length equals the radius of its circumcircle. This makes it easier to construct because the same method can be used whether given the side length or the circumradius.

  1. Draw a circle with radius r equal to the side length and a diameter AB. Mark point D on the circle.
  2. With center A and radius AO, draw an arc that intersects the circle at points B and F. Similarly, with center D and radius DO, draw an arc that intersects the circle at points E and C.
  3. Connect these points to form the hexagon.

Razado the Regular Heptagon Knowing its Side

  1. Draw side AB and draw a perpendicular at one end, for example, B. Also, draw the bisector of this side.
  2. At point A, construct a 30° angle with AB, extending the side until it intersects the perpendicular from B at point P. Use a bevel to measure the 30° angle.
  3. With center A and radius AP, draw an arc that intersects the bisector of AB. (Missing information)

Tangencies

Two figures are tangent if they have one point in common, known as the point of contact. The harmonious union between curves and straight lines or curves with each other is called tangency, and the union must occur by contact. The most common tangencies in geometric designs are those generated between lines and circles, and between circles themselves.

Basic Properties of Tangencies

To determine the exact paths of tangencies, consider the following theorems:

  • First Theorem: A line is tangent to a circle when they have only one point (M) in common, and the line is perpendicular to the radius of the circle at that point.
  • Second Theorem: A circle is tangent to two intersecting lines if its center is located on the bisector of the angle between the lines.
  • Third Theorem: Two circles are tangent if they have a common point (N) that is collinear with the centers of the circles.

Tracing a Tangent Line to a Circle Knowing a Point P on the Line

  1. Draw the radius connecting the center O to point P.
  2. Draw a line perpendicular to the radius through point P. This line is the desired tangent r.

Tracing Tangents to a Circle from an External Point P

  1. Connect point P with the center of the circle, O, and draw the perpendicular bisector of segment OP, obtaining point H.
  2. With radius HO and center H, draw an arc that intersects the given circle at points M and M’, which are the points of contact.
  3. Connect point P with M and M’ to draw the tangent lines r and s.

Drawing a Circle of Known Radius Tangent to Two Converging Lines

  1. Draw the perpendicular bisector of the angle formed by the two lines.
  2. Draw a line parallel to one of the given lines and separated from it by the known radius r. The intersection of this line with the bisector is the center of the circle to be drawn.
  3. The tangent points are M and M’, which are found by drawing radii perpendicular to lines r and s.

Drawing a Circle Through Point A and P that is Tangent to Line r

  1. Since M and P have to be points on the circle, its center must lie on the bisector of MP.
  2. As P is the point of tangency to line r, the center O of the circle is located where the perpendicular from P to MP intersects the bisector.

Drawing External Tangents to Two Circles of Different Radii

  1. Connect the centers O and O’ and find the midpoint of OO’, which we call H.
  2. Draw a circle concentric to the larger circle with a radius equal to the difference between the larger and smaller radii.
  3. With center H and radius HO, draw an arc that intersects the auxiliary circle at M and M’.
  4. Connect O to M and M’, resulting in points U and V.
  5. Draw two radii from O’ parallel to OU and OV to obtain points S and T. Connect V with T and U with S to draw the tangent lines r and s.

Drawing Internal Tangents to Two Circles of Different Radii

  1. Connect the centers O and O’ and determine the midpoint of OO’, which is H.
  2. Draw a circle with radius r + r’ and center O.
  3. Draw another circle with radius HO and center H, which intersects the former at points M and M’.
  4. Connect points M and M’ with O to obtain points U and V.

Tracing a Circle of Radius r Tangent to Another Circle Outside with Center at P

  1. Extend a radius from O containing point P.
  2. Add the radius r from P to obtain O’.
  3. Finally, draw a circle with center O’ and radius O’P.

Drawing a Circle Tangent to Another Circle at a Point M and Passing Through Another Interior Point N

  1. Since M and N are points on the same circle, the center will be on the bisector of MN.
  2. Connect M with O, which intersects the bisector, to obtain the center O’ of the desired circle. Draw the circle with radius O’N.

Drawing a Circle of Known Radius r, Tangent to Another Circle and a Given Line

  1. Draw an arc with center O and radius equal to the sum of the radius of the given circle and the known radius r.
  2. Draw a line parallel to the given line and separated from it by the known radius r. The intersection of this arc with the parallel line is the center O’ of the desired circle. Points M and N are the points of contact.