- where a pair of interior opposite angles are both
- with one side and one diagonal having equal lengths.
For today’s purpose, let’s give the pseudonym special to cyclic quadrilaterals having the above two properties, and then prove that in any such special quadrilateral:
connecting the side-lengths of a quadrilateral with the diagonals and the distance between the midpoints of the diagonals.
Consider the right kite shown below, with longer diagonal , and shorter diagonal .
Since in the kite shown above and because it’s a right kite, Euler’s quadrilateral formula (2) gives
This example shows that apart from right kites, there are other (cyclic) quadrilaterals that satisfy equation (1), including the ones with the pseudonym special, which is our focus for today.
Such a special quadrilateral contains two interior opposite angles measuring ; place them at and .
If we let diagonal , then , and so by Euler’s quadrilateral formula (2):
Using the same diagram/notation for the previous example, we have:
Since one of the diagonals has the same length as one of the sides, it can’t be diagonal because is the diameter. So let’s suppose diagonal as the same length as side . Then the above equation gives .
Diagonal will now have the same length as sides and .
Suppose that . Let and be the circumcenter and orthocenter of the parent triangle . Then the quadrilateral is a particularly pleasant right kite.
- (Special property) Let be special, as per the definition at the introduction; specifically, let the interior opposite angles at and be both , and let diagonal have the same length as side . PROVE that:
- the radius through is parallel to side
- the orthocenter of triangle is a reflection of vertex over side .