Math 444 Class 10/28

Inscribed Quadrilaterals

In the figure below, the arcs have angle measure a1, a2, a3, a4.

Write down the angle measures of the vertex angles of the quadrilateral:

Angle DAB __________

Angle ABC __________

Angle BCD __________

Angle CDA __________

Now compute the sum of the opposite vertex angles.  You can compute this sum not only in terms of letter, but you can find a numerical answer.  What is it?

Angle DAB + Angle BCD =

Angle ABC + Angle CDA =

For the quadrilaterals ABCD below, the quadrilateral cannot be inscribed in a circle.  When the circle through A, B, C is constructed, the vertex D is not on the circle.  In each case the point E is defined by intersecting ray CD with the circle.

From the preceding work, tell what is the sum of angle ABC + angle CEA?

Using this and facts about angles, what can you show is true about the sum

Angle ABC + angle CDA in each of the cases?

Summary of Results:

Theorem: A quadrilateral ABCD can be inscribed in a circle if and only if a pair of opposite angles is supplementary.

Comment: It is true that one pair of supplementary angles is supplementary  if and only if both pairs are supplementary, since the sum of all the angles is 360 degrees.

Proof:  In the quadrilateral ABCD can be inscribed in a circle, then we have seen above using the inscribed angle theorem that the sum of either pair of opposite angles = (1/2(a1 + a2 + a3 + a4) = (1/2)360 = 180.

Conversely, if the quadrilateral cannot be inscribed, this means that D is not on the circumcircle of ABC.  On the second page we saw that this means that the sum of angle B + angle D is either greater than or less than 180 degrees.

Angle Locus Theorem:  Given two points P and Q and a fixed angle measure a, then the set of points R on one side of line PQ so that angle PRQ = a is an arc with endpoints PQ and central angle 360 - 2a (excluding P and Q from the set of R).

Comment: When a = 90 degrees, this is the Carpenter Theorem.

Comment:  This means that the set of all points R with PRQ = a is the union of two arcs (with P and Q removed), one on each side of the line PQ.  As an example is the set of points R so that PRQ = 60 degrees.