Difference between revisions of "Van Aubel's Theorem"
(→Proof 1: Complex Numbers) |
(→Proof 1: Complex Numbers) |
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\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.