Interactive step-by-step animation using GeoGebra. Post your solution in the comment box below.

Level: Mathematics Education, High School, Honors Geometry, College.

Details: Click on the figure below.

## Saturday, February 29, 2020

### Geometry Problem 1455: Nagel Point, Excircles, Incircle, Congruent Segments

Labels:
congruence,
dynamic geometry,
excircle,
GeoGebra,
Geometry,
HTML5,
incircle,
ipad,
ipadpro,
Nagel theorem,
tangency point

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Using Van Aubel's theorem

ReplyDeleteWe get (AN/NTa)=(ATc/TcB)+(ATb/TbC)=((s-b)/(s-a))+ ((s-c)/(s-a))

AN/NTa= a/(s-a), adding 1 to both sides we get ATa/NTa=s/(s-a)

Or NTa/ATa=(s-a)/s ........(1)

If we draw a tangent to incircle which is parallel to BC, it will touch incircl at Ia. Lets assume altitude from A to BC is h. Then we get

AIa/ATa=(h-2r)/h, replacing h by 2S/a and r by S/s, we get

AIa/ATa= (s-a)/s .....(2)

Using eq (1) and (2) we have AIa= NTa

Similary BIb=NTb and CIc= NTc

https://photos.app.goo.gl/vEqkL8Da7ogdieyr5

ReplyDeleteLet a, b, c are 3 sides of triangle ABC and 2p is its perimeter

Define point J, R(A) and r per attached sketch

From I(A) draw B’C’// BC

Note that AB’C’ is the image of ABC in the homothetic transformation

Since circle J is the excircle of triangle ABC => circle I is the excircle of triangle AB’C’

So AI(A)/ AT(A)= r/R(A)= (p-a)/p…..(1)

Apply Van Aubel II theorem in triangle ABC with point N

we have AN/AT(A)= (p-b)/(p-a) + (p-c)/(p-a)= a/(p-a)

From above expression we will have NT(A)/AT(A)= (p-a)/p…..(2)

Compare (1) to (2) we have AI(A)=NT(A)

Similarly we will have same results with other 2 vertexes