Предмет: Геометрия, автор: mironova1915

в треугольнике авс точка х лежит на стороне ав а точка у лежит на стороне вс. отрезки ау и сх пересекаются в точке з. известно , ау = су и ав=сз. докажите что точки в х з лежат на одной окружности.

siestarjoki: Три точки всегда лежат на одной окружности. Понятно, что спрашивают про четыре точки.

Ответы

Автор ответа: siestarjoki

Докажем, что точки B, X, Y, Z лежат на одной окружности.

△AYC - равнобедренный.

Ось симметрии р/б треугольника - серединный перпендикуляр к основанию (m).

Возьмем точку B1 симметричную B относительно m.

∠B=∠B1, AB=CB1 (по построению)

AB=CZ=CB1 => △B1CZ - р/б => ∠B1=∠CZB1

=> ∠CZB1=∠B

Внешний угол четырехугольника BXYZ равен противолежащему внутреннему - четырехугольник вписанный (вершины лежат на одной окружности).

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