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HomepageCullen's Parallelogram
Jim Cullen


Elementary Geometry & the Parallelogram Law


Basic construction of Cullen's Parallelogram In Euclidean geometry, a parallelogram is defined as a quadrilateral with two pairs of parallel sides, implying through the parallel postulate congruent opposite sides and angles. In Fig 1 is Cullen's Parallelogram ABCD with the vertices, sides, and two main diagonals labelled. Notice that the lengths of all the sides and the two main diagonals are all integer values. This is the first quality that makes the parallelogram special. By the Parallelogram Law, the sums of the squares of the four sides is equal to the sums of the squares of the two main diagonals. In this case:

2 · (2890)2 + 2 · (4760)2 = 42502 + 66302 = 62019400


Cullen's Parallelogram was identified in a computer search for parallelograms with specific characteristics, characteristics involving integer values for various dimensions including the radius of a circle circumscribed about three points of the parallelogram. As is turns out, Cullen's Parallelogram is actually the tenth integer multiple of the smallest primitive example of a 'magic' or all integer parallelogram. The primitive has sides equal to 289 and 476, and main diagonals equal to 425 and 663, such that:


2 · (289)2 + 2 · (476)2 = 4252 + 6632 = 620194


If the sides are are ordered from smaller to larger and called A and B, then for this parallelogram A=289 and B=476. The diagonals are likewise ordered and so C=425 and D=663. Refer to the following diagram for the remaining integer dimensions.

Basic parallelogram.


As you can see from the diagram in Fig. 1a, the height is the vertical height of the parallelogram, labelled as H. Shown also is V, which is the overall width of the parallelogram that extends beyond the length of B. If the parallelogram is stood up on its shorter side, then AH and AV are the alternate height and alternate width. For the smallest primitive all integer parallelogram, H=255, V=136, AH=420, and AV=224, all of them integer values. Magic parallelograms have integer values for all these described dimensions. Cullen's Parallelogram is a Magic Parallelogram that also allows a circle with an integer radius to be circumscribed about the parallelogram, with sides A and B arranged as secants of the circumscribed circle. The following table provides the data on the primitive Magic Parallelograms with side A less than 10000 and side B limited to less than 20000. There are a total of just twelve primitive Magic Parallelograms in this range. The data provided also includes the area of the parallelogram and the radius of the circumscribed circle.


ABCDHAHVAVAreaRadius
289476425663255420136224121380375.7
5004375422545754684095176154020475002443.910256
9571682118924656601160693121811101201787.125
162525352210364014002184825128735490002112.5
1869792172988900168071208193471133072804950.625
2555532937967446168035041925401589527205662.0625
36635476551375113465518011881776189743403970.1
549195377922133964560792030595313434887208065.508333
57461127177351613336407140444687214102644012733.54643
712418769152072397554601438545761205610247874015640.83333
84051025043051824534444200766793503530100022263.24405
991111236103351849784159540523659369455094010892.67778


Notice from the table that the next largest Cullen Parallelogram would be the second integer multiple of the fourth primitive Magic Parallelogram which has the following dimensions: A=3250, B=5070, C=4420, D=7280, H=2800, AH=4368, V=1650, and AV=2574. The area of this parallelogram is 14196000 and the radius of the circumscribed circle is 4225. The search continues for primitive Magic Parallelograms, the goal being to uncover a primitive that is also a Cullen Parallelogram; the radius of it's circumscribed circle will be an integer in the above table. So far, all Cullen Parallelograms are integer multiples of one of the primitives.



Altitudes & Circumscribed Circle


Altitudes and Circumscribed Circle In Fig. 2, Cullen's Parallelogram is shown with the altitudes of the parallelogram, and a circle circumscribed through vertices A, B, and C. Altitude BE, which is the height of the parallelogram, is 2550 which is an integer value. The alternate height CF, also an integer, measures 4200. These altitudes form right triangles ( BED & CFD ) along either side of the parallelogram, with all three sides of both triangles having integer lengths. Two points have been added to the figure. Point P is the center of the parallelogram, located at the intersection of the two main diagonals. Point O is the center of a circle circumscribed through the points A, B, and C. The radius of the circle also has an integer value, equal to 3757. The integer heights and integer radius of the circumscribed circle are the other two properties of Cullen's Parallelogram that make it special, at least as far as parallelograms go. From here it will be noticed that the distance between points O and P is an integer value of 1768 and forms a right triangle also with all integer valued sides. It will also be noticed that the distance between points E and F is an integer length of 3120, forming the basis for another parallelogram with all integer components obeying the parallelogram law; its main diagonals measure 3120 and 2000.



Impressive Cluster of Right Triangles


Impressive Cluster of Right Triangles Here we examine the right triangle formed by the horizontal and vertical components of the segment OP separating the centers of the parallelogram and the circumscribed circle. The circumscribed circle about Cullen's Parallelogram has a radius 3757, and the center of the circle is point O. The center of the parallelogram, at the intersection of the two main diagonals, is marked as point P. The distance between points O and P is an integer value 1768, shown as the green dashed segment OP in Fig. 3. Three temporary points, a, b, and c, have been added to the diagram. Segment OP is the hypotenuse of the right triangle OPa. The lengths of the other two sides of this right triangle are: Pa = 680 and Oa = 1632, such that 17682 = 6802 + 16322. Note that segment Oa is the sum of the lengths of the segments Oc=357, cb=765, and ca=510, or 357 + 765 + 510 = 1632. If additional segments are constructed between point P and points b and c, more integer sided right triangles are formed, including right triangle Dcb. This hypotenuse is equal to 1275 with sides equalling 765 and 1020, such that 12752 = 7652 + 10202. Notice that segment Db = 1275 is equal to segment ac = 510 + 765 = 1275.



Summary of the Properties of the Circumscribed Circle


Property Summary of Circumscribed Circle Here are given the basic measurements of the circumscribed circle about Cullen's Parallelogram, in relationship to the elements of the parallelogram, many of which measure again to integer values. If we were to display segments such as OA, OB, or OC, they would measure the integer value 3757 or the radius of the circumscribed circle, since points A, B, and C all lie on the circle. Temporary points c and a have been added to the diagram, the bisectors of segments CA and AB respectively. Segments Od and Ob are segments extended from O, through the bisectors d and b, and out to where those lines intersect the arc of the circle. The resulting segments cd = 289 and ab = 850, between the bisectors and the arc of the circle, are integer values.


There are many interesting relationships between Cullen's Parallelogram and it's circumscribed circle. In Fig. 4, consider the right triangle Ocf. The hypotenuse is segment Oc with a length given as 3468. Since point c bisects segment CA, one leg of this right triangle is formed by segment cf, equal to the sum of the lengths of segments cP = 4760 / 2 = 2380 and Pf = 680, or cf = 2380 + 680 = 3060. The other leg of this right triangle is formed by segment Of, equal to the sum of the lengths of segments Oe = 357 and ef = 1275, or Of = 357 + 1275 = 1632. The Pythagorean Theorem then shows that 34682 = 30602 + 16322 = 12027024. Another right triangle may be formed by extending segment Oc to the arc of the circle so that the new hypotenuse is equal to Od = 3757. This multiplies the former right triangle by a factor of 13/12. The new horizontal leg is then segment dg = 3315 and the new vertical leg is segment Og = 1768, or the same as the length OP. The new Pythagorean relationship is then 37572 = 33152 + 17682 = 14115049. Expanding on this, we have point P which bisects segment CB, resulting in two segments CP and PB each of length 3315. The length of segment OP is given as 1768. The definition of a circle gives the lengths of segments OC and OB as 3757, meaning that triangles OPC and OPB are congruent right triangles.


Other Relationships Within the Circumscribed Circle


Other Relationships Within the Circumscribed Circle Cullen's Parallelogram ABCD is shown again in Fig. 5, within it's complete circumscribed circle. The green portions of the diagram show the right triangles associated with the parallelogram's altitudes and alternate altitudes. The yellow section shows the smaller parallelogram DEFG that is just 8/17 the size of the original but identical in shape. The relationship here, again described by the Parallelogram Law, is; 2 · (1360)2 + 2 · (2240)2 = 20002 + 31202 = 13734400. The light blue section shows the congruent back to back right triangles that completes the diagram to the opposite side of the circumscribed circle. This arrangement is not the only possible one but is the one originally found to contain points J, K, and L, that are located on the radius of the circumscribed circle. One very interesting relationship not shown on this diagram is the fact that points A, O, G, and L, are all collinear, and this line passes through the centerpoint of the circle O. The length of segment OG ( not shown ) would then also have an integer length, 2093, and the length of segment AG ( not shown ) would have an integer length of 5858.



An Alternate Set of Relationships Within the Circumscribed Circle


Alternate Relationships Within the Circumscribed Circle Cullen's Parallelogram ABCD is shown with an alternate set of relationships in Fig. 6. Several triangles in this diagram are similar ( same shape but different size ). Triangles PFE, AKJ, CKD, and BDJ, are all similar isosceles triangles, with the ratio of the length of one of the two equal sides to the length of the remaining side being equal to 17/16. Triangles LBC and LFE are similar scalene triangles. In both, the ratios of the two longer sides to the shortest sides are 65/57 and 68/57.


If you find any other relationships of interest in any of these diagrams, please feel free to contact me at the email link given below. This document is in progress...

The diagrams shown in this article were generated with the help of GeoGebra, an online Java geometry application that may also be downloaded for free from their website. This is a very intuitive and powerful application that makes quick work of geometric constructions, the measurements of the elements, and the generation of exported graphics of the completed construction. I recommend GeoGebra highly.


URL: http://members.bex.net/jtcullen515/math9.htm
© 2010: Jim Cullen - all rights reserved.
Contact the author with any questions or comments: Jim Cullen
Last Update: October 31, 2010.

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