Problem-Solving and Selected Topics in Euclidean Geometry by Sotirios E. Louridas & Michael Th. Rassias

Author:Sotirios E. Louridas & Michael Th. Rassias
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

Prove that the ratio of the perimeter of the triangle A 1 B 1 C 1 to the area of the triangle A 1 B 1 C 1 is constant.

Solution

We know that for any triangle, the ratio of its area to its semi-perimeter is equal to the radius of its inscribed circle (it is left as an exercise for the reader).

We shall therefore show that the circle inscribed in the triangle has a constant radius (see Fig. 6.5).

Fig. 6.5Illustration of Problem 6.1.4

We observe that the points A,A 1,K and C,C 1,K are collinear when K is the intersection of the bisectors of the triangle A 1 B 1 C 1.

This is the case because A is the intersection of the bisectors of GTA, where T is the intersection of the lines AB,l,l 3.

In this case, K lies on the bisector AA 1 and C is the intersection of the bisector of with the bisector of the exterior angle , where L is the intersection of the lines BC,l,l 1.

Therefore, the line CC 1 is the bisector of the exterior angle when G is the intersection of AC,l,l 2. This means that the point K belongs to the bisector CC 1.

We observe that

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.