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

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

Details: Click on the figure below.

## Monday, December 10, 2018

### Geometry Problem 1407: Right Triangle, Incircle, Excircle, Collinear Tangency Points, Collinearity

Labels:
collinear,
excircle,
geometry problem,
incircle,
right triangle,
tangency point

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Let BC touch in circle at X & AC at Y.

ReplyDeleteLet AB touch excircle at Z.

Let BD = p & FX = q

BDIX is a square of side p & BFEZ is a square of side p+q

Now DZ = 2p+q = YG = q + 2.CF, so CF = CG = p

Hence Tr. s BDF & GCE are congruent SAS and hence < BFD = < CEG = < CFG since CFEG is concyclic

It follows that D,F,G are collinear

Sumith Peiris

Moratuwa

Sri Lanka

Draw DIP diameter. Ang APD = ACB = 2.CGF => PDG = CGF

ReplyDeletebut PDG = DF'B => DF'B = CFG (CF=CG equal tangencial seg)