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midpoint wanted, concurrent cevians given (Kyiv City Olympiad 2008 10.4)

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June 30, 2020

geometrymidpointCevians

Problem Statement

Given a triangle

ABC

,

A {{A} _ {1}}

,

B {{B} _ {1}}

,

C {{C} _ {1}}

- its chevians intersecting at one point.

{{A} _ {0}}, {{C} _ {0}}

- the midpoint of the sides

BC

and

AB

respectively. Lines

{{B} _ {1}} {{C} _ {1}}

,

{{B} _ {1}} {{A} _ {1}}

and

{ {B} _ {1}} B

intersect the line

{{A} _ {0}} {{C} _ {0}}

at points

{{C} _ {2}}

,

{{A} _ {2}}

and

{{B} _ {2}}

, respectively. Prove that the point

{{B} _ {2}}

is the midpoint of the segment

{{A} _ {2}} {{C} _ {2}}

.
(Eugene Bilokopitov)

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