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```
Date: 05/06/99 at 07:57:30
From: Jayant
Subject: Permutations and Combinations

Ten sticks of lengths 1,2,3,....,10 are given. Four are selected to
so formed. Also find in how many of these quadrilaterals a circle can
be inscribed.
```

```
Date: 05/07/99 at 11:19:20
From: Doctor Anthony
Subject: Re: Permutations and Combinations

The condition for a quadrilateral ABCD to have an inscribed circle is
that

AB + CD = AD + BC

You can prove this easily by considering that the tangent to the
circle from point A along AB is equal to the length of the tangent
along AD, similarly the tangents from B are equal along BA and BC, and
so on with tangents from C along CB and CD and tangents from D along
DC and DA. Adding up these equal lengths leads to the result quoted
above.

So the four sticks must satisfy the condition that the sum of one pair
is equal to the sum of the other pair

So    1+10 = 2+9 = 3+8 = 4+7 = 5+6

This gives:                               C(5,2) = 10 quadrilaterals

Similarly 2+10 = 3+9 = 4+8 = 5+7  giving  C(4,2) = 6 quadrilaterals

"  3+10 = 4+9 = 5+8 = 6+7       "     C(4,2) = 6       "

"  4+10 = 5+9 = 6+8             "     C(3,2) = 3       "

"  5+10 = 6+9 = 7+8             "     C(3,2) = 3       "

"  6+10 = 7+9                   "     C(2,2) = 1       "

"  7+10 = 8+9                   "     C(2,2) = 1       "

"  1+9 = 2+8 = 3+7 = 4+6        "     C(4,2) = 6       "

"  1+8 = 2+7 = 3+6 = 4+5        "     C(4,2) = 6       "

"  1+7 = 2+6 = 3+5              "     C(3,2) = 3       "

"  1+6 = 2+5 = 3+4              "     C(3,2) = 3       "

"  1+5 = 2+4                    "     C(2,2) = 1       "

"  1+4 = 2+3                    "     C(2,2) = 1       "
--------------------------

Since the order must stay as opposite pairs, we could keep one pair
fixed and swap over the other pair giving double this answer, but if
we can view the quadrilateral from either side, we would divide by 2
again, so we stick at 50 quadrilaterals which can have an inscribed
circle.

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Permutations and Combinations
High School Triangles and Other Polygons

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