Difference between revisions of "1977 USAMO Problems/Problem 4"
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== Problem == | == Problem == | ||
Prove that if the opposite sides of a skew (non-planar) quadrilateral are congruent, then the line joining the midpoints of the two diagonals is perpendicular to these diagonals, and conversely, if the line joining the midpoints of the two diagonals of a skew quadrilateral is perpendicular to these diagonals, then the opposite sides of the quadrilateral are congruent. | Prove that if the opposite sides of a skew (non-planar) quadrilateral are congruent, then the line joining the midpoints of the two diagonals is perpendicular to these diagonals, and conversely, if the line joining the midpoints of the two diagonals of a skew quadrilateral is perpendicular to these diagonals, then the opposite sides of the quadrilateral are congruent. | ||
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+ | ==Hint== | ||
+ | Skew quadrilateral, perpendicular? Use vectors! | ||
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+ | |||
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== Solution == | == Solution == | ||
− | {{ | + | We first prove that if the opposite sides are congruent, then the line joining the midpoints of the two diagonals is perpendicular to these diagonals. |
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+ | Let the vertices of the quadrilateral be A, B, C, D, in that order. Thus, the opposite sides congruent condition translates to (after squaring): | ||
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+ | <cmath>AB^2 = CD^2, BC = AD^2</cmath> | ||
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+ | Let the position vectors of A, B, C, D be a, b, c, d with respect to an arbitrary origin <math>O</math>. Define the square of a vector <math>a^2 = a * a</math>. Thus, <math>(a-b)^2 = (c-d)^2</math> and so <cmath>a^2 - 2ab + b^2 = c^2 - 2cd + d^2.</cmath> Similarly, <cmath>b^2 - 2bc + c^2 = a^2 - 2ad + d^2.</cmath> Subtracting the second equation from the first eventually results in <cmath>a^2 - ab + cd + bc - ad - c^2 = 0,</cmath> which factors as <cmath>(a - b + c - d)(a - c) = 0.</cmath> | ||
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+ | Let M and N be the midpoints of <math>AC</math> and <math>BD</math>, with position vectors <math>m = \frac{a+c}{2} and n = \frac{b+d}{2}</math>, respectively. Then notice that the perpendicularity condition for vectors gives <math>MN</math> perpendicular to <math>AC</math>. Similarly (by adding the equations), we find that <math>MN</math> is perpendicular to <math>BD</math>, completing the first part of the proof. | ||
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+ | Proving the converse is straightforward also: indeed, <math>MN</math> perpendicular to both diagonals gives | ||
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+ | <cmath>a^2 - ab + cd + bc - ad - c^2 = 0</cmath> | ||
+ | <cmath>b^2 - ab + cd - bc + ad - d^2 = 0</cmath> | ||
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+ | Adding and subtracting the equations eventually simplifies to | ||
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+ | <cmath>(a-b)^2 = (c-d)^2</cmath> | ||
+ | <cmath>(a-d)^2 = (b-c)^2,</cmath> | ||
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+ | or <math>AB^2 = CD^2</math> and <math>AD^2 = BC^2</math>. This reduces to <math>AB = CD</math> and <math>AD = BC</math>, as desired. <math>\blacksquare</math> | ||
== See Also == | == See Also == |
Revision as of 15:10, 14 August 2014
Contents
[hide]Problem
Prove that if the opposite sides of a skew (non-planar) quadrilateral are congruent, then the line joining the midpoints of the two diagonals is perpendicular to these diagonals, and conversely, if the line joining the midpoints of the two diagonals of a skew quadrilateral is perpendicular to these diagonals, then the opposite sides of the quadrilateral are congruent.
Hint
Skew quadrilateral, perpendicular? Use vectors!
Solution
We first prove that if the opposite sides are congruent, then the line joining the midpoints of the two diagonals is perpendicular to these diagonals.
Let the vertices of the quadrilateral be A, B, C, D, in that order. Thus, the opposite sides congruent condition translates to (after squaring):
Let the position vectors of A, B, C, D be a, b, c, d with respect to an arbitrary origin . Define the square of a vector . Thus, and so Similarly, Subtracting the second equation from the first eventually results in which factors as
Let M and N be the midpoints of and , with position vectors , respectively. Then notice that the perpendicularity condition for vectors gives perpendicular to . Similarly (by adding the equations), we find that is perpendicular to , completing the first part of the proof.
Proving the converse is straightforward also: indeed, perpendicular to both diagonals gives
Adding and subtracting the equations eventually simplifies to
or and . This reduces to and , as desired.
See Also
1977 USAMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.