Geometry Problem. Post your solution in the comment box below.

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

Details: Click on the figure below.

## Tuesday, December 11, 2018

### Geometry Problem 1408: Right Triangle, Incircle, Excircle, Incenter, Midpoint, Tangency Point, Collinearity

Labels:
collinear,
excenter,
excircle,
incenter,
incircle,
midpoint,
right triangle,
tangency point

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let incircle radius r, touch BC at H;

ReplyDeleteexcircle radius R,

Right ABC, the length is a,b,c, c>a,c>b;

r=(a+b-c)/2; R=(c+a-b)/2;

if triangle IHF similar to triangle BDF, then D,I F colinear,

it easy to show r/(b/2) =(R-r)/R

so D,I,F colinear.

If X & Y are the tangency points of the in circle at BC & BA & if the radii of the incircle is p & of the excircle is q & S(ABC) = S, s the semi perimeter,

ReplyDeleteXF/IX = (q-p)/p = (a-2p) / p = (a-2S/s)/(S/s)

= (as - ac)/(ac/2) = (s-c)/(c/2) = BF/BD

So Tr.s BDF & XIF are similar and DIF are collinear

Sumith Peiris

Moratuwa

Sri Lanka