Question

In the proof of Lemma 1 we mentioned that many incorrect methods for finding a vertex such that the line segment is an interior diagonal of have been published. This exercise presents some of the incorrect ways has been chosen in these proofs. Show, by considering one of the polygons drawn here, that for each of these choices of , the line segment is not necessarily an interior diagonal of .

a) p is the vertex of P such that the angle$\mathbf{\angle }\mathbf{abp}$is smallest.

b) p is the vertex of P with the least -coordinate (other than ).

c) p is the vertex of P that is closest to .

Short answer

(a) The line segmentbp is not necessarily an interior diagonal ofP as it lies outside of the polygon.

(b) The line segmentbp is not necessarily an interior diagonal ofP as it lies outside of the polygon.

(c) The line segmentbp is not necessarily an interior diagonal ofP as it lies outside of the polygon.

Step-by-step solution

Identification of given data

The given data can be listed below as:

• The name of the line segment is .
• The name of the vertex is p.
• The name of the polygon is P .

Significance of the polygon 

A polygon is described as a plane figure which consist of three angles and straight sides. Moreover, a polygon can also contain more than five sides also.

(a) Determination of the first choice of p

The diagram of the polygon P has been drawn below:

Here, the point P is described as the vertex of the polygon P where is the smallest angle. Taking an example, in the above diagram, all the angles such as $\angle abq,\angle abr,\angle abs$and$\angle abc$ are greater than the angle $\angle abp$. Moreover , which is the line segment, is not considered as the polygon’s interior diagonal, as the line segment’s part lies outside the polygon.

Thus, the line segmentbp is not necessarily an interior diagonal of P as it lies outside of the polygon.

(b) Determination of the second choice of p

The diagram of the polygonp has been drawn below:

Here, the point p is described as the vertex of the polygonp that consists of the smallest coordinate of $x$, which is other than the point$b$ . The$x$ coordinate is the left-most point of the polygon. Taking an example, in the above diagram, the left most point is other than the point $b$. Moreover $bd$, which is the line segment, is not considered as the polygon’s interior diagonal, as the line segment’s part lies outside the polygon.

Thus, the line segment$bp$ is not necessarily an interior diagonal ofp as it lies outside of the polygon.

(c) Determination of the third choice of p

The diagram of the polygon has been drawn below:

Here, the pointp is described as the vertex of the polygon that is closest to the point b. Taking an example, in the above diagram, the left most point is other than the point b and both the points are close to each other. Moreover bd, which is the line segment, is not considered as the polygon’s interior diagonal, as the line segment’s part lies outside the polygon.

Thus, the line segment bp is not necessarily an interior diagonal p of as it lies outside of the polygon.