Difference between revisions of "Van Aubel's Theorem"
m |
(→Proof 1: Complex Numbers) |
||
(8 intermediate revisions by 2 users not shown) | |||
Line 1: | Line 1: | ||
= Theorem = | = Theorem = | ||
− | On each side of quadrilateral <math>ABCD</math>, construct an external square and its center: | + | On each side of quadrilateral <math>ABCD</math>, construct an external square and its center: <math>ABA'B'</math>, <math>BCB'C'</math>, <math>CDC'D'</math>, <math>DAD'A'</math>; yielding centers <math>P_{AB}, P_{BC}, P_{CD}, P_{DA}</math>. Van Aubel's Theorem states that the two line segments connecting opposite centers are perpendicular and equal length: |
− | < | + | <math>P_{AB}P_{CD} = P_{BC}P_{DA}</math>, and <math>\overline{P_{AB}P_{CD}} \perp \overline{P_{BC}P_{DA}}</math>. |
= Proofs = | = Proofs = | ||
== Proof 1: Complex Numbers== | == Proof 1: Complex Numbers== | ||
− | Putting the diagram on the complex plane, let any point < | + | <asy> |
+ | size(220); | ||
+ | import TrigMacros; | ||
+ | rr_cartesian_axes(-3,8,-2,8,complexplane=true,usegrid = false); | ||
+ | pair A, B, C, D, O, P, Q, R, SS; | ||
+ | O = (0,0) ; | ||
+ | A = (2,1.5); | ||
+ | B= (4,1.8); | ||
+ | C = (5.3,3); | ||
+ | D= (3,5.3); | ||
+ | |||
+ | draw(A--B--C--D--cycle); | ||
+ | draw(A--(A + rotate(-90)*(B-A))--(B + rotate(90)*(A-B))--B); | ||
+ | draw(B--(B + rotate(-90)*(C-B))--(C + rotate(90)*(B-C))--C); | ||
+ | draw(C--(C + rotate(-90)*(D-C))--(D + rotate(90)*(C-D))--D); | ||
+ | draw(D--(D + rotate(-90)*(A-D))--(A + rotate(90)*(D-A))--A); | ||
+ | |||
+ | P = (B + (A + rotate(-90)*(B-A)))/2; | ||
+ | Q = (C + (B + rotate(-90)*(C-B)))/2; | ||
+ | R = (D + (C + rotate(-90)*(D-C)))/2; | ||
+ | SS = (A + (D + rotate(-90)*(A-D)))/2; | ||
+ | |||
+ | //draw(WW--Y,red); | ||
+ | //draw(X--Z,blue); | ||
+ | dot("$a$",A,SW); | ||
+ | dot("$b$",B,2*E); | ||
+ | dot("$c$",C,E); | ||
+ | dot("$d$",D,NNW); | ||
+ | |||
+ | dot("$p$",P,E); | ||
+ | dot("$q$",Q,S); | ||
+ | dot("$r$",R,N); | ||
+ | dot("$s$",SS,S); | ||
+ | </asy> | ||
+ | Putting the diagram on the complex plane, let any point <math>X</math> be represented by the complex number <math>x</math>. Note that <math>\angle PAB = \frac{\pi}{4}</math> and that <math>PA = \frac{\sqrt{2}}{2}AB</math>, and similarly for the other sides of the quadrilateral. Then we have | ||
Line 27: | Line 61: | ||
\end{eqnarray*}</cmath> | \end{eqnarray*}</cmath> | ||
− | Finally, we have < | + | Finally, we have <math>(p-r) = i(q-s) = e^{i \pi/2}(q-s)</math>, which implies <math>PR = QS</math> and <math>PR \perp QS</math>, as desired. |
+ | |||
==See Also== | ==See Also== | ||
[[Category:Theorems]] | [[Category:Theorems]] |
Latest revision as of 14:01, 4 March 2023
Theorem
On each side of quadrilateral , construct an external square and its center:
,
,
,
; yielding centers
. Van Aubel's Theorem states that the two line segments connecting opposite centers are perpendicular and equal length:
, and
.
Proofs
Proof 1: Complex Numbers
Putting the diagram on the complex plane, let any point
be represented by the complex number
. Note that
and that
, and similarly for the other sides of the quadrilateral. Then we have
From this, we find that
Similarly,
Finally, we have , which implies
and
, as desired.