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Geometry Mathley 7. 2 IK / IO= r / R

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June 7, 2020

circumcircleincircleequal ratio

Problem Statement

A non-equilateral triangle

D,E, F

is inscribed in a circle

\Gamma

with centre

O

, radius

R

and its incircle has centre

I

and touches

BC,CA,AB

at

D,E, F

, respectively. A circle with centre

I

and radius

r

intersects the rays

[ID), [IE), [IF)

at

A',B',C'

. Show that the orthocentre

K

of

\vartriangle A'B'C'

is on the line

OI

and that

\frac{IK}{IO}=\frac{r}{R}

Michel Bataille .

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