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Geometrical Constructions 1 Geometrical Constructions 1 by Pál Ledneczki Ph.D. Geometrical Constructions 1 „Let no one destitute of geometry enter my doors” (Plato, 427-347 BC) RAFFAELLO: School of Athens, Plato & Aristotle Budapest University of Technology and Economics ? Faculty of Architecture ? Department of Architectural Representation 2005 Fall Semester 2 © Pál Ledneczki Geometrical Constructions 1 Drawing Instruments For note taking and constructions For drawings (assignments) - loose-leaf white papers of the size A4 - three A2 sheets of drawing paper (594 mm × 420 mm) - pencils of 0.3 (H) and 0.5 (2B) - set of drawing pens (0.2, 0.4, 0.7) - eraser, white, soft - set square (two triangles, 30 cm) - set square (two triangles, 20 cm) - A1 sheet of drawing paper for folder - pair of compasses with joint for pen frame 10 folder - French-curves (title box on drawings) - set of color pencils 15 I -1 NAME (A1 size of 15 Date Signature GR # - drawing board paper folded into two) 30 30 60 30 lettering Name Architectural Representation Budapest University of Technology and Economics ? Faculty of Architecture ? Department of Architectural Representation 2005 Fall Semester 3 © Pál Ledneczki Geometrical Constructions 1 Contents 1) Basic constructions 2) Loci problems 3) Geometrical transformations, symmetries 4) Pencils of circles, Apollonian problems 5) Regular polygons, golden ratio 6) Affine mapping, axial affinity 7) Conic sections Budapest University of Technology and Economics ? Faculty of Architecture ? Department of Architectural Representation 2005 Fall Semester 4 © Pál Ledneczki Geometrical Constructions 1 1 Basic Constructions Use of Set Squares and Pair of Compasses copying angle bisecting an angle perpendicular bisector A B of a segment parallel lines by perpendicular sliding ruler line by rotating ruler Budapest University of Technology and Economics ? Faculty of Architecture ? Department of Architectural Representation 2005 Fall Semester 5 © Pál Ledneczki Geometrical Constructions 1 1 Basic Constructions Triangles, Some Properties C Triangle inequalities ? a a+b > c, b+c > a, c+a > b b Sum of interior angles ß a+? a B a + ß + ?= 180° c A Any exterior angle of a triangle is equal to the sum of the two interior angles non adjacent to the given exterior angle. About the sides and opposite angles, if and only if b > a than ß > a for any pair of sides and angles opposite the sides. Classification of triangles; acute, obtuse, right, isosceles, equilateral Budapest University of Technology and Economics ? Faculty of Architecture ? Department of Architectural Representation 2005 Fall Semester 6 © Pál Ledneczki Geometrical Constructions 1 1 Basic Constructions Theorems on Right Triangle Theorem of Pythagoras In a right-angled triangle, the sum of the squares b a of the legs is equal to the square of the hypotenuse. c Converse: If the sum of the squares of two sides in a triangle a2 + b2 = c2 is equal to the square of the third side, then the triangle is a right triangle. C Theorem of Thales In a circle, if we connect two endpoints of a diameter with any point of the circle, we get a right A B angle. Converse: If a segment AB subtends a right angle at the point C, then C is a point on the circle with diameter AB. Budapest University of Technology and Economics ? Faculty of Architecture ? Department of Architectural Representation 2005 Fall Semester 7 © Pál Ledneczki Geometrical Constructions 1 1 Basic Constructions Triangles, Special Points, Lines and Circles O The three altitudes of a triangle meet at a point O. This point is the orthocenter of the triangle. The orthocenter of a right triangle is the vertex at the right angle. The three medians of a triangle meet at a point G (point of gravity). This G point is the centroid of the triangle. The centroid trisect the median such that the segment connecting the vertex and the centroid is the 2/3-rd of the corresponding median. C The three perpendicular bisectors of the sides of a triangle meet at a point C. This point is the center of the circumscribed circle of the triangle. e G O The three points O, G, and C are collinear. The line passing through them is C the Euler’s line. The ratio of the segments OG and GC is equal to 2 to 1. The three bisectors of angles of a triangle meet at a point P. This point is the center of the inscribed circle. (Two exterior bisectors of angles and the P interior bisector of the third angle also meet at a point. About this point a circle tangent to the lines of the triangle can be drawn.) Budapest University of Technology and Economics ? Faculty of Architecture ? Department of Architectural Representation 2005 Fall Semester 8 © Pál Ledneczki Geometrical Constructions 1 1 Basic Constructions Theorems on Circles Circles are similar Two circles are always similar. Except the case of congruent or concentric circles, two circles have two C2 centers of similitude C1 and C2. C1 C 2 Theorem on the angles at the circumference and at the C1 center The angle at the center is the double the angle at the " circumference on the same arc. " " Corollaries: 2" A chord subtends equal angles at the points on the A " same of the two arcs determined by the chord. B In a cyclic quadrilateral, the sum of the opposite angles is 180°. B1 A P 1 Circle power A2 On secants of a circle passing through a point P, the B2 product of segments is equal to the square of the 2 2 2 2 2 T tangential segment: PA1 +PB1 =PA2 +PB2 =PT Budapest University of Technology and Economics ? Faculty of Architecture ? Department of Architectural Representation 2005 Fall Semester 9 © Pál Ledneczki Geometrical Constructions 1 1 Basic Constructions Parallel Transversals, Division of a Segment Theorem on parallel transversals: the ratios of the y corresponding segments of arms of an angle cut by parallel x lines are equal. x’ y’ x:y = x’:y’, x:x’ = y:y’ Divide the segment AB by a point C such that AC:CB = p:q. A u Find the point C. p=3 l Solution: 1) draw an arbitrary ray r from A C q=5 2) measure an arbitrary unit u from A onto the ray p + q times, get the point B’ B’ 3) connect B and B’ r B 4) draw parallel l to BB’ through the endpoint of segment p 5) C is the point of intersection of l and AB Budapest University of Technology and Economics ? Faculty of Architecture ? Department of Architectural Representation 2005 Fall Semester 10 © Pál Ledneczki Geometrical Constructions 1 1 Basic Constructions Ratios of Segments in Triangle In a right triangle, h 2 pq, b2 cp, a2 cq. a c = = = b hc The altitude assigned to the hypotenuse is the geometrical mean of the two segments of the hypotenuse. p q c A leg is the geometrical mean of the hypotenuse and the orthogonal projection of the leg on the hypotenuse. In an arbitrary triangle, B q a p b = . q c p c C The bisector of an angle in a triangle divides the A b c opposite side at the ratio of the adjacent sides. Budapest University of Technology and Economics ? Faculty of Architecture ? Department of Architectural Representation 2005 Fall Semester 11 © Pál Ledneczki Geometrical Constructions 1 1 Basic Constructions Quadrilaterals Cyclic quadrilateral Circumscribed quadrilateral Diagonals and bimedians a b " d M $ c " +$ =180° a +c = b + d coincidence of midpoints Kite (Deltoid) Trapezium Rhombus a a a a Budapest University of Technology and Economics ? Faculty of Architecture ? Department of Architectural Representation 2005 Fall Semester 12 © Pál Ledneczki Geometrical Constructions 1 1 Basic Constructions Euclid of Alexandria (about 325BC-265BC) More than one thousand editions of The Elements have been published since it was first printed in 1482. BL van der Waerden assesses the importance of the Elements: ‘Almost from the time of its writing and lasting almost to the present, the Elements has exerted a continuous and major influence on human affairs. It was the primary source of geometric reasoning, theorems, and methods at least until the advent of non-Euclidean geometry in the 19th century. It is sometimes said that, next to the Bible, the "Elements" may be the most translated, published, and studied of all the books produced in the Western world.’ Quotations by Euclid: This is a detail from the fresco The School of Athens by Raphael There is no royal road to geometry. A youth who had begun to read geometry with Euclid, when he had learnt the first proposition, inquired, "What do I get by learning these things?" So Euclid called a slave and said "Give him threepence, since he must make a gain out of what he learns." That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles. [the 5th postulate] http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Euclid.html Budapest University of Technology and Economics ? Faculty of Architecture ? Department of Architectural Representation 2005 Fall Semester 13 © Pál Ledneczki Geometrical Constructions 1 1 Basic Constructions Geometrical Constructions Euclidean construction (instruments: straight edge and a pair of compasses) : We may fit a ruler to two given points and draw a straight line passing through them We may measure the distance of two points by compass and draw a circle about a given point We may determine the point of intersection of two straight lines We may determine the points of intersection of a straight line and a circle We may determine the points of intersection of two circles Formal solution of constructions problems: Sketch Draw a sketch diagram as if the problem was solved Plan Try to find relations between the given data and the unknown elements, make a plan Algorithm Write down the algorithm of the solution Discussion Analyze the problem in point of view of conditions of solvability and the number of solutions Budapest University of Technology and Economics ? Faculty of Architecture ? Department of Architectural Representation 2005 Fall Semester 14 © Pál Ledneczki Geometrical Constructions 1 How to Solve It? (according to George Polya, 1887 - 1985) First.
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