The “analysis” method forconstruction problems in thedynamic geometryHee-chan Lew
Korea National University of Education
SEMEO-RECSAM University of Tsukuba of Tsukuba
Joint SeminarFeb. 15, 2016, Tokyo

Key word• Analysis:– A working backward strategy to find aconstruction or deductive proof methods ingeometric problems– Pappus, the Greek mathematician in AD 3rd
century, systemized in his book “TheCollections”– It has not been used after his era, in myopinion, because of the lack of drawing tools
to operate the method efficiently.

Conclusion• DG is a very good environment for
students to find a construction method bythemselves for geometric construction
problems and to prove that it is a proper
method deductively.
• DG is an environment to revive theanalysis method of Pappus who criticized
Euclid’s deductive method so called“synthesis”

Euclid “Elements”• In BC 3rd century Euclid wrote thegreat books so called “Elements” basedon mathematical activities of 300 years
from BC 6th to BC 3rd century• It has been a typical geometry textbookof mankind for over 2000 years

Euclid “Elements”• It was written by an axiomatic anddeductive method• It has been used as an unique textbook
to develop students’ deductive proof
ability since Greek era.

Euclid “Elements”• It was edited as a textbook for a small
group of elite students at the time of BC
3rd Century.
• It might be not proper for ordinary
students, like in nowadays, who dislike
mathematics seriously, particularly
dislike proof.

Educational Defect• “Elements” does not show the processof mathematical discovery as follows:– imagination,– experimentation,– reasonable guess,– analogy,– trial and error,– (sometimes) mistakes and failure etc

Educational Defect• It gives only final results by mathematicians
without showing the behind story.
• It makes normal teachers easy to explain theproof process appeared in the books one byone• It is difficult for students to get the reason
why the particular proof method was selected.
– It makes students think mathematics a very
difficult subject

New Direction for Euclidean
Geometry• Students must reconstruct theconstruction process and design its
proving process by themselves rather
than absorbing the proceeses fromteachers.
• Students must think independently andproductively.

 Intuition, guessing, investigation, measuring,and trial and error.

New Method• It is almost impossible in Paper andPencil environment.
• Furthermore DG does not guarantee thesuccess.
• We need a special method in DG toimprove deductive proof abilities ofordinary students.

“Analysis”• The analysis can provide an alternative

teaching method for Euclidean geometry

which is very difficult to normal students• The Oldest strategy among themathematics heuristics• Pythagorean school also emphasized theanalysis but, in “Elements” not appeared.
• In AD 3rd century, Pappus systemized it in
his famous book, “the collections”.

Definition of Analysis• In analysis, we assume that which issought as if it were already done and weinquire what it is from which this results
and again until we come up somethingalready known or belonging to the class offirst principles

Definition of Synthesis• In synthesis, reversing the analysis, wetake as already done that which was last
arrived at in the analysis and we arrive
finally at the construction of what wassoughtDialectic unity• Greek thought the dialectic integration ofanalysis and synthesis as a substance ofmathematical thought. However, Euclid’s
Elements considered the synthesis to
reduce theorems from the foundation as away to guarantee the truth ofmathematics.

Example 1
Prove that every triangle has one inscribedcircle.

Analysis
Assume that triangle ABC has one inscribed circle.
Triangle ABC has one circle contacted at one point of each
three sides of the triangle
Triangle ABC has one point D from which the distances tothree sides of the triangle are same.
This D is an intersection of two sets; 1) the set of points
whose distance from AB and AC are same and 2) the setof points whose distance from CB and CA.
B

This is an intersection of
the bisector of angle A andthe bisector of angle C.

DASynthesis
Draw an angle bisector of angle BAC and BCA
The intersection
point D is such thatthe distances fromthree sides of thetriangle are same.

BEDF

ED = DG = DF
There is one
inscribed circle DAGCExample 2
There is a circle O andthere are two lines mand n which areperpendicular each otherlike the following figure.
Construct a circle whose
center is located on theline n and to which thecircle O and the line m
are tangentnOm

Analysis
Assumethat thecircle P isconstructedsatisfying
the givenconditions.

nOPm

Analysis
Draw a circle P with radius OP
Let F is anintersection point of
the line n and thecircle
The triangle is anisosceles.
Draw a line
perpendicular to OF
passing through P
E is a mid point of
OFnOPEmF

Synthesis
Draw a circle H with radius OG
Let F is theintersection pointn

Draw aperpendicularbisector of OF
P is an intersectionof theperpendicularbisector and line nDraw a circle Pwith radius PHOPGEHmF

Problem 1: Tangent line
Draw a tangent line of Circle A through point B
BA

Analysis
Assumethat theline BC is atangent
line of thecircle A

BCAAnalysis
Drawsegments ACand AB
Triangle ABC isa right triangle
Let D is a
midpoint of thesegment AB
DC = DA = DBBCDA

Synthesis
Draw a segment
AB and let D is a
midpoint of AB
Draw a circle Dwith radius DA andlet C is aintersection pointof the two circles
AC is the tangent
line to findBCDA

Problem 2: The Shortest path
Someone wants to move from A to B via one
point on the seashore through the shortest
path. Find the point on the seashore.

Analysis
Determine the point C such that the sum of two segments
is least by using length and calculation function inmeasurement menu and by dragging a point C.
Assume that the point C is what we want to find.

Analysis
Draw a line AC
Draw aperpendicular line
BD to theseashore CETwo triangles CBEand CED arecongruent
BE = EDSynthesis
Draw a perpendicularline BE
Draw a circle E withthe radius BE
D is an intersection
point of the circle andthe line
Draw a line AD
C is the point to findProblem 3: Fermat point
In the interior partof triangle ABC,
find a point F suchthat the sum of thelength of threesegments AF, BF,
CF is least.
We call the point
Fermat point.

AFBCAnalysis

Determine the point F such that the sum of threesegments is least by using length and calculation function
in measurement menu and by dragging a point F.
Assume that the point is the Fermat point.
A
FA = 3.0427 cm
FB = 3.2949 cm
FC = 4.3518 cm
FA+FB+FC = 10.6894 cm

FB

CAnalysis
Construct a circle F, I, J of which radii are AF, BF,
CF respectively.
Two triangles AHB, AGC are equilateral.
GA

H
I

J
F

B
C

Synthesis
Construct equilateral triangles AHB and AGC
Draw lines HC and GB
The intersection point F is the Fermat point.
GHA

FBC• Please construct a point the sum of distances tothe following points A and B is same as thegiven segment as below.

ABAnalysis

Synthesis

Problem 5: Hypobola
Please construct a point the difference ofdistances to the following points A and B issame as the given segment as below.

ABAnalysis

Synthesis

Problem 6: Parabola• Please construct a point from which distances
to point A and the below line is same.

AAnalysis

Synthesis

DG as a tool for analysis•

DG can help students draw a precise
figure.

• To find more easily a series of necessary
conditions of final conclusion, studentshave to show relations among components
in the problem situation.
• In a figure roughly drawn on the paper it isvery difficult for students to find therelation.

DG as a tool of analysis•

DG has a measure function.

– It makes students determine the goodstarting point for analysis by measuring
length or angle etc continuously whiledragging the point continuously.
– In paper circumstance, it is very difficult for
students to make a precise measuring
enough to find the good starting point foranalysis.

DG as a tool of analysis• DG is dynamic– It can make student perform various
experiment to find necessary conditions bydrawing, erasing and manipulating figureseasily as well as dynamically.
– In paper and pencil circumstance, it is almost
impossible to perform analysis because thefigure drawn on the paper cannot bemanipulated.

DG as a tool of analysis•DG is a reflective tool

• If the relation among components ispreserved while dragging the picture
constructed by synthesis, the construction
process can be considered as a rightprocedure.
• In paper and pencil circumstance, there is noway to check whether the construction
process is right or not.

Conclusion• In the late of 1980s, Cabri and GSP were
designed as a dynamic tool for students toinvestigate the properties and relation
within and between figures throughoperating figures on the computer screen
directly.

Conclusion•

More than traditional construction bynormal compasses and ruler– Construct, erase, drag and transform figures– Measure segments and angles•

DG: A dynamic method for Euclidean
geometry proposed by Clairaut, Frenchmathematician in the 17th century– But, then he also had not the proper toolConclusion• In DG, students can make a conjecture togeometric properties and confirm theminformally and feel the need to prove theconjectured and informally confirmed
geometric facts.
• In DG, students can improve their proof
abilities by using the analysis method.

Conclusion• DG is a very excellent tool for the analysis
method which is a good mathematicalstrategy proposed by Greek
mathematicians but forgotten for a longtime because of maybe a lack of propertool.

Conclusion• In mathematics textbooks, the conclusion
is a conclusion.
• In mathematics education, the conclusion
should be a starting point rather than aconclusion.
• DG can provide a better and safe routefrom the starting point to the developmentof deductive proof of normal students.

Suggestion• There need to be various LS experiment inwhich students can actually perform
analysis well by using DG.
• We have to investigate students’ thoughtprocess of analysis in various construction
problems.