Eisatopon Math AI Challenges: A Characterization of the Euler Line

Δευτέρα 1 Μαΐου 2017

Let

V_{1} V_{2} V_{3}

be a triangle with circumcenter

O

, and let

M_{1}, M_{2}, M_{3}

be the midpoints of

V_{2} V_{3}, V_{1} V_{3}

, and

V_{1} V_{2}

, respectively.

For

- \infty < t \leq \infty

and

k = 1, 2, 3,

let Mtk be the point defined by

O M t_{k} = t O M_{k}

, (where by

M t_{\infty}

we mean the point at infinity in the direction of

O M_{k}

Prove that for any

t \in (- \infty, \infty]

, the lines

V_{k} M t_{k}, k = 1, 2, 3,

are concurrent, and that the locus of all such points of concurrence is the Euler Line of

Δ V_{1} V_{2} V_{3}