Geometry Problems

GEOMETRY PROBLEMS

A triangle's incentral triangle is the triangle formed by the points of intersection of the angle bisectors with the sides. In the above diagram, DEF is the incentral triangle ABC.

1. When I calculated the inradii of the nth incentral triangle, [i.e. the incentral triangle of the 
(n-1)th incentral triangle, and so on] I noticed quite an interesting pattern- for ANY isosceles ABC, the
nth inradius is almost half of the (n-1)th. 

 By looking at the graphs below, we get an idea of how close to 1/2 the reduction in inradii is.

For an isosceles triangle with a = 12, and k = 18: graph of the inradius of nth incentral triangle VS n (distinct points connected by curve)

Graph of (exponent is x+4 instead of x to start the graph from a value as close to the 1st inradius as possible.)

Devise a function to express the decrease for the nth incentral triangle- where - and prove that the reduction in inradii will keep decreasing (tending to 1/2).
  

2.   Given a triangle, find:
a. an equation for the position of the nth incentral triangle’s incenter with respect to the given
triangle’s incenter I. In addition, check if the motion of all incenters will always be oscillatory
about I.

b. An inequality among the areas of  the triangle, its incircle, circumcircle, and nine-point circle.

3. Given a circle with radius R:

a. If R is prime, find the number of inscribed/circumscribed X-sided polygons with prime sides.

b. Find all inscribed/circumscribed triangles that have the same area A.

4.   

A given conic section intersects an N-sided polygon in some arbitrary manner. Find the


area of the intersection. It fascinates me that although this problem seems very simple at first sight, it actually is not!

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