Problem 2

A non-isosceles triangle

A_1A_2A_3

has sides

a_1

a_2

a_3

with the side

a_i

lying opposite to the vertex

A_i

. Let

M_i

be the midpoint of the side

a_i

, and let

T_i

be the point where the inscribed circle of triangle

A_1A_2A_3

touches the side

a_i

. Denote by

S_i

the reflection of the point

T_i

in the interior angle bisector of the angle

A_i

. Prove that the lines

M_1S_1

M_2S_2

and

M_3S_3

are concurrent.

Step 8 of 8: The homothety center is the point of concurrency

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In plain words

This is the payoff of setting up a homothety in the first place: concurrency through the center is automatic, not something that needs a separate argument.

M_1S_1,\ M_2S_2,\ M_3S_3 \text{ concur at the center } P \text{ of the homothety}

Detailed analysis

By the defining property of a homothety with center $P$ , every point and its image are collinear with $P$ : since $M_i\mapsto S_i$ for $i=1,2,3$ , each line $M_iS_i$ passes through $P$ . Hence $M_1S_1$ , $M_2S_2$ , $M_3S_3$ all pass through $P$ , i.e. they are concurrent, as required.