Useful Lemma in Geometry

Here is a very useful document of in-center and ex-center lemma for solving mathematical problems, especially Olympiad Geometry.

Thanks Evan Chen (IMO Gold Medalist) for this document.
Click Here for PDF File !


Two Parallel Lines Meet at Infinity!

Conventional geometry is called Euclidean geometry. There are certain properties which define it but it has an inconsistency which even the ancient Greeks who formulated it knew about. One
properties is that any two distinct lines define a point (i.e. where they intersect), unless the lines are parallel. In contrast, any two distinct points define a line, with no exceptions. Projective geometry copes with these two cases symmetrically: no exceptions. The aspects of projective geometry, in essence, are :

  • Any two distinct points define a line (i.e. no exception of two parallel lines)
  • Any two distinct lines intersect at a point
We mentioned earlier that projective geometry can cope with vanishing points, corresponding
to parallel lines “meeting” at infinity. This might seem a strange claim, given that all we have
done is add a scale factor to Euclidean geometry. In fact there is another way of describing
projective geometry which brings out this aspect better. Start by thinking of the 4D coordinates
like this: the (x, y, z) part defines a direction and the w part defines the depth or scale along that
direction. Now suppose we take all the finite points, which correspond to Euclidean geometry,
and add points at infinity in all possible directions. The result is projective geometry!
(Honestly, I also have no idea about the intersection of two parallel lines. I have never ever believed it, and the only one time I recognize this aspect is during my Olympiad Math Exam).

The Projective Geometry will be a useful place to explore more about this topic. Projective Geometry

Here is the review of Projective Geometry :

Review Projective Geometry


Spiral Similarity

A spiral similarity about a point O (known as the center of the spiral similarity) is a composition​of​a​​ rotation and a dilation, both centered at O. (See diagram)


Fact: Let A;B;C;D be four distinct point in the plane such that ABCD is not a parallelogram. Then
there exists a unique spiral similarity that sends A to B, and C to D.

Lemma:  Let A;B;C;D be four distinct point in the plane, such that AC is not parallel to BD. Let
lines AC and BD meet at X. Let the circumcircles of ABX and CDX meet again at O. Then O is the
center of the unique spiral similarity that carries A to C and B to D.


Fact. If O is the center of the spiral similarity that sends A to C and B to D, then O is also the center of the spiral similarity that sends A to B and C to D.


Learning How to Count

  • Multiplication: Consider that an experiment consists of a sequence of two procedures say, A and B. Let NA and NB denote the number of ways in which one can execute A and B, respectively. It then follows that there is N=NA*NB ways of executing such an experiment. In general, if the experiment consists of a sequence of k procedures, then one may run it in N=N1*N2*….*Nk different ways.
  • Addition: Suppose now that the experiment involves k procedures in parallel(rather than in sequence). This means that we either execute the procedure 1 or the procedure 2 or….. or procedure k. If Ni denote the number of ways that one may carry out the procedure i={1,2,3….,k}, then there are N=N1+N2+…..+Nk ways of running such an experiment.
  • Permutation: Suppose now that we have a set of N different elements and we wish to know the number of sequences we can construct containing each element once, and only once. Note that the concept of sequence is distinct from that of a set, in that order of appearance matters. For instance, the sample space {a,b,c} allows for the following permutations (abc,acb,bac,bca,cab,cba). In general, there are N! possible permutations out of N elements because there are N options for first element of the sequence, but only N-1 options for the second element, and so on until we have only one option for the last element of the sequence. There is also a more general meaning for permutation in combinatorics for which we form sequences of k different elements from a set of N elements. This means that we have N options for the first element of the sequence, but then N-1 options for the second element and so on until we have only N-k+1 options for the last element of the sequence. It thus follows that we have N!/(N-k)! permutations of k out of N elements in this broader sense.
  • Combination: This is a notion that only differs from permutation in that ordering does not matter. This means that we just wish to know how many subsets of k elements we can construct out of a set of N elements. For instance, it is possible to form the following subsets with two elements of {a,b,c,d}: {a,b};{a,c};{a,d};{b,c};{b,d};{c,d}. Note that {b,a} is not counted because it is exactly the same subset as {a,b}. This suggests that, in general, the number of combinations is inferior to the number of permutations because one must be counted only one of the sequences that employ the same elements but with a different ordering. In view that there are N!/[(N-k)!k!] possible combinations of k out of N elements.


Source: Statistics for Business and Economics (Marcelo Fernandes)

For grade 11-12

Enjoy This Paper


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