Dilation of Incircle to Excircle - Simulated | Math Olympiad Geometry | Diameter of Incircle Lemma скачать в хорошем качестве

Dilation of Incircle to Excircle - Simulated | Math Olympiad Geometry | Diameter of Incircle Lemma

https://www.cheenta.com/diameter-of-i... Suppose we have a triangle ABC. Let us extend the sides BA and BC. We will draw the incircle of this triangle. Here is the construction. Draw any two angle bisectors, say of angle A and angle B Mark the intersection point I. Drop a perpendicular line from I to one of the sides, say AC in this picture. Suppose the perpendicular intersects AC at E. The incircle is drawn centered at I and with radius IE Suppose EI intersects the incircle at F Now let us draw the excircle To do that we will need the angle bisector of external angle A and external angle C Suppose they intersect at I_A. Drop a perpendicular from I_A to extended BA or extended BC or AC. In this picture we drop it on extended BA Suppose J is the point of intersection of extended BA and the perpendicular Draw a circle centered at I_A and radius I_AJ The incircle can be dilated or blown up with respect to point B into the excircle. The center I is sent to the center I_A under dilation FE which is perpendicular to AC is sent to another segment perpendicular to AC as angles are preserved under dilation Questions: What is the ratio of dilation? How can you rigorously show that the center I goes to the center I_A under this dilation Where do the point F go under this dilation Show that AN = CE Problem useful for I.S.I B.Stat B.Math Entrance, CMI Entrance and Math Olympiad