Download How to work out a proof on your own

MS2013: EUCLIDEAN GEOMETRY. HOW TO BUILD PROOFS.
ANCA MUSTATA
1. How to construct a proof
We’ll work with the following example:
Let ABC be an isosceles triangle with |AB| = |AC|. Consider the point
M on the segment [AB] and the point N on the segment [AC]. Let D denote
the intersection of BN with CM. Show that AD is the angle bisector of
[ if and only if |BM| = |CN|.
the angle BAC
Step 1). Read the text and draw a picture. Avoid putting new features in
the picture which are not in the text! Thus, avoid any of the following three
sketches:
A
A
b
b
b
A
b
M
b
N
b
M
b
b
b
M
b
N
N
B
b
C
b
B
b
C
b
B
b
C
because the triangle ABC is not assumed equilateral, nor are M or N assumed to
be midpoints of [AB] or [AC], nor are BN or CM assumed to be bisectors of B̂ and
Ĉ!
Rather, try one or two dissimilar sketches:
A
A
b
b
b
M
b
b
M
b
b
B
D
b
b
N
D
N
b
C
b
B
b
C
Step 2) Make sure you understand all the mathematical terms in the text. Review
[
relevant definitions if necessary. Thus, ”AD is the angle bisector of the angle BAC”
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ANCA MUSTATA
\ = DAC,
\ while ”if and only if” means that you will need to solve
means that BAD
two different problems. Set these up right away.
Problem (1):
Assume: △ ABC isosceles with |AB| = |AC|.
\=\
BAD
DAC.
To Prove: |BM| = |CN|.
Problem (2):
Assume: △ ABC isosceles with |AB| = |AC|.
|BM| = |CN|.
\=\
To Prove: BAD
DAC.
For the next steps I will work with Problem (2). Try Problem (1) on your own.
Step 3) Form a general strategy based on the type of knowns/unknowns and
available theorems. At this step you may want to consult the list (or logic scheme)
of main theorems, propositions and corollaries proven so far.
Since the problem is about equal segments/lines, the general strategy here will
be to use congruent triangles.
Step 4) Start from the conclusion and look backwards, trying to gather up the
data you will need in order to prove it. Keep an eye on the assumptions to see if some
of them provide the data you need, but keep in mind that you might not be able to
complete your proof in just one step.
Problem (2):
Assume: △ABC isosceles with |AB| = |AC|.
|BM| = |CN|.
\=\
To Prove: BAD
DAC.
\ = DAC.
\ Your plan is to use congruent triangles. You have two
You want BAD
options:
(1) Try to prove the congruence of the triangles ADB and ADC. Here we know
|AB| = |AC| and AD is a common side, but we will certainly won’t be able to
use the angles in between these sides, because their equality is what we need
to prove. So no SAS. We might be able to use SSS if we prove |BD| = |CD|.
(2) Try to prove the congruence of the triangles ADM and ADN. Here we know
AD is a common side, and we can deduce |AM| = |AN| by difference of
equal segments because |AB| = |AC| and |BM| = |CN|. Again, we know
nothing of the angles so far, so we may want to use SSS like before. We’ll
need |DM| = |DN|.
At this point, the two options look quite similar. I have a preference for option (1)
because the segments |BD| and |CD| are more tied up with the triangle ABC, which
is one of our main assets, as we know it’s isosceles!
How can we prove |BD| = |CD|? We have two options: congruent triangles, or
show that △DBC is isosceles. By the theorem on isosceles triangles, this is equivalent
\ = NBC.
\ Keep this on the to do list!
to proving MCB
Step 5) Take a deeper look at the assumptions, review the theorems involved.
Make sure we employ all assumptions:
Assumption 1: △ABC isosceles with |AB| = |AC|. By the theorem on isosceles
[ = ACB.
[
triangles, this also implies ABC
MS2013: EUCLIDEAN GEOMETRY. HOW TO BUILD PROOFS.
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Assumption 2: |BM| = |CN|.
Look at the assumptions 1 and 2 together, (mark them out in a picture). We note
BC is a common side and so we have △MBC ≡ △NCB by SAS. That’s perfect
\ = NBC
\ from our to do list! Now we go backwards through
’cause it proves MCB
step 4) and prove |BD| = |CD|, hence the congruence of the triangles ADB and
\=\
ADC, hence BAD
DAC.
You note that there were other trails that I left out cold. The reason was that I
didn’t know much about the elements involved. You need to analyze things carefully
to find which trails are the best to take, but if in doubt, try each and see which gets
you to the finish line first.
2. Problem solving tips:
• Know your theory. All the theorem, lemmas, propositions in the Lecture notes,
as well as a good part of the Exercise Set 1, are basic facts which you should know if
you wish to find proofs easily.
• Whenever you have to prove something about a geometric object, start by analyzing what is known about that objects’ parts. (For example, if the object is
an angle, check what is known about its sides. Is one side perpendicular on another
line? Then your angle might be part of a right angled triangle...etc. If the object is
a segment: how have the end-points been defined?...)
You may also find it useful to split an angle into a sum of two angles, or a
segment into a sum of two segments, and check what is known about the
summands.
As well, you may want to include your object into more complex structures,
like e.g. triangles, and check what you know about the elements (sides,
angles) of those structures (triangles).
• Make a list of given data (Assumptions), and what is to be proven (Conclusion).
Start from the Conclusion and think of possible proof strategies based on the type of
data you have. For example, if the assumptions comprise many perpendicular lines
or angle bisectors or points on circles, your strategy should probably comprise angle
calculations. If your assumptions deal with lengths of segments, possible strategies
may use theorems like Pytagoras’, Thales, Menelaus, Ceva or the power of a point.
Or, you may have mixed strategies using both angles and segment calculations, like
congruent or similar triangles, symmetries, etc.
Some strategies for proving that three points A, B, C are collinear:
a) (Angle-based strategies) Choose a line d in your diagram which passes
through one of the points A, such that the other two points B and C are on
different sides of this line. Prove that the angles formed by d with AB and
[
AC are equal, and so they must be opposite angles. Or, split the angle BAC
◦
[ = 180 . Or, prove
into a sum of angles and prove by computations that BAC
that both AB and AC are parallel to the same line.
b) (Segment-based strategy) Use a corollary of Thales’s theorem (see the
notes on area) or Menelaus’s theorem (more rarely used).
c) (Mixed strategy). Choose two of the points BC. Does the line BC have any
special property? Is it an angle or perpendicular bisector, a median, midline,
perpendicular or another line or simply the reflection of another line? Then
prove that point A also has the universal property satisfied by all the points of
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ANCA MUSTATA
BC. For example, in the problem above, the points A, D and the midpoint of
BC are collinear. You can prove this by first noticing that the line connecting
A with the midpoint of BC is the perpendicular bisector of segment BC,
because △ABC is isosceles. Then showing that |DB| = |DC|, you prove D
must also be on the perpendicular bisector of BC (from the universal property
of the perpendicular bisector).
Some strategies for proving that three lines a, b, c are concurrent:
a) Intersect two of the lines and prove that the point of intersection is collinear to
two special points on the third line. Use one of the strategies for collinearity.
b) Let a and b intersect at X, let a and c intersect at X ′ . Choose a special point
A on a and prove that X and X ′ are equally distanced from, and on the same
side of A.
c) If the lines are in the interior of a given triangle you may try to use Ceva’s
theorem. If the lines are each perpendicular on some other line, use Pytagoras’
theorem like in the alternate proof of the existence of the orthocentre. For
example, you may use this method to prove that the common chords of three
intersecting circles are concurrent.
• Whenever you use one of the hypotheses given in the Assumptions (”We know”)
section of our Problem, put a check mark besides it. If you ever become stuck in
a proof, go back to the assumption list and look for those without a check mark.
Perhaps it’s time you use them. At the end of a proof, check if all the assumptions
have been used. If that is not the case, it is possible that there is an error in the
proof (or perhaps you’ve used an assumption without noticing).
• It is often useful to choose a ”reference object”. For example, if your problem
contains a triangle ABC, then you can choose this triangle as your reference. Calculate all the angles of interest to you in terms of the angles A, B and C. This
will allow you to easily compare different angles. Similarly, calculate the segments
of interest to you in terms of the sides a, b, c of the triangle.
If your problem contains a circle C, calculate the angles of interest to you
in terms of arcs on the circle. This will allow you to spot relations between
various angles which may not be obvious otherwise. As well, you can move easily
between properties of arcs and properties of segments, as equal arcs correspond to
equal chords and vice-versa. Calculate segments in terms of the radius and distances
of points from the centre.
• Don’t be afraid to add new elements to your diagram. Routinely connect
points not yet connected if they seem relevant to the problem.
When adding new lines, it’s often useful to mimic the type of constructions already
in the statement of the problem: if midpoints are given, you may want to extend
some segments by equal parts to get new midpoints. If some points are symmetric
with respect to a line, draw the symmetric images of other points with respect to
that line. If interested in comparing lengths of segments, you may draw some parallel
lines...
• If no other method works, you can solve your problem for a special choice
of one of the points in the diagram with no special properties. For example, if you
cannot solve your problem when D is a random point on the line BC, you can try
to solve the case when D is the midpoint first. Then imagine D slowly sliding on
BC away from the midpoint (draw a few diagrams). Which features of your problem
change? How do they compare with the special case? Can you calculate the general
MS2013: EUCLIDEAN GEOMETRY. HOW TO BUILD PROOFS.
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case in rapport to the special case? Important note: This method should be
avoided unless the special case really helps you calculate the general case.
Solving a special case of in itself is worth very little. Moreover, if the
point D already has some special property given in the Assumptions part
of the problem, then adding new properties alters the problem, which is
a serious mistake. For example, if AD is stated to be the altitude of the triangle
△ABC, do not assume D to also be the midpoint of BC! This would mean you
assume △ABC to be isosceles, which is wrong unless the problem explicitly states
△ABC to be isosceles.
3. How to study a proof
Rule no.1: Don’t memorize proofs! Read them. Analyze all steps, i.e. find the
answer to the following questions:
• What are we proving at this step?
• Why is this step necessary?
• Which assumptions/theorems are we using and why?
Find and memorize the punch-line/central idea!
After reading the proof, let it be for some time, then look again at the problem.
Does the punch-line come naturally to you? Can you reconstruct the proof from the
punch-line? If not, analyze the proof again. etc.