Problem 1

Let

ABC

be a triangle and

J

the center of the

A

-excircle. This excircle is tangent to the side

BC

M

, and to the lines

AB

and

AC

K

and

L

, respectively. The lines

LM

and

BJ

meet at

F

, and the lines

KM

and

CJ

meet at

G

. Let

S

be the point of intersection of the lines

AF

and

BC

, and let

T

be the point of intersection of the lines

AG

and

BC

. Prove that

M

is the midpoint of

ST

Step 2 of 5: Write the key points in barycentric coordinates

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In plain words

A point dividing a side (or its extension) in a known ratio has barycentric coordinates equal to the opposite ratio of lengths, and the A-excenter has the standard coordinates (-a:b:c).

M=(0:s-b:s-c),\ K=(-(s-c):s:0),\ L=(-(s-b):0:s),\ J=(-a:b:c)

Detailed analysis

In homogeneous barycentric coordinates with respect to $\triangle ABC$ : $M$ divides segment $BC$ with $BM=s-c,CM=s-b$ , giving $M=(0:s-b:s-c)$ ; $K$ lies on ray $AB$ beyond $B$ with $AK=s,BK=s-c$ , giving $K=(-(s-c):s:0)$ ; similarly $L=(-(s-b):0:s)$ ; and the $A$ -excenter has the standard coordinates $J=(-a:b:c)$ . Altogether, $M=(0:s-b:s-c),\ K=(-(s-c):s:0),\ L=(-(s-b):0:s),\ J=(-a:b:c)$ .