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angle bisector intersects circumcircle at X

Source: Bulgaria 1965 P3

June 23, 2021

geometryangle bisectorcircumcircleTriangles

Problem Statement

In the triangle

ABC

, angle bisector

CD

intersects the circumcircle of

ABC

at the point

K

.
(a) Prove the equalities:

\frac1{ID}-\frac1{IK}=\frac1{CI},\enspace\frac{CI}{ID}-\frac{ID}{DK}=1

where

I

is the center of the inscribed circle of triangle

ABC

. (b) On the segment

CK

some point

P

is chosen whose projections on

AC,BC,AB

respectively are

P_1,P_2,P_3

. The lines

PP_3

and

P_1P_2

intersect at a point

M

. Find the locus of

M

when

P

moves around segment

CK

.

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