Difference between revisions of "Van Aubel's Theorem"
(→Proof 1: Complex Numbers) |
(→Proof 1: Complex Numbers) |
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draw(B--(B + rotate(-90)*(C-B))--(C + rotate(90)*(B-C))--C); | 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(C--(C + rotate(-90)*(D-C))--(D + rotate(90)*(C-D))--D); | ||
− | draw(D--(D + rotate(-90)*(A-D))--(A + rotate(90)*(D-A))-- | + | draw(D--(D + rotate(-90)*(A-D))--(A + rotate(90)*(D-A))--A); |
P = (B + (A + rotate(-90)*(B-A)))/2; | P = (B + (A + rotate(-90)*(B-A)))/2; | ||
Line 30: | Line 30: | ||
//draw(WW--Y,red); | //draw(WW--Y,red); | ||
//draw(X--Z,blue); | //draw(X--Z,blue); | ||
− | dot("$a$", | + | dot("$a$",A,SW); |
− | dot("$b$", | + | dot("$b$",B,2*E); |
− | dot("$c$", | + | dot("$c$",C,E); |
− | dot("$d$", | + | dot("$d$",D,NNW); |
dot("$p$",P,E); | dot("$p$",P,E); | ||
Line 61: | Line 61: | ||
\end{eqnarray*}</cmath> | \end{eqnarray*}</cmath> | ||
− | Finally, we have <math>(p-r) = i(q-s) = e^{i \pi/2}(q- | + | 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.