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Romania NMO 2023 Grade 7 P3

Source: Romania National Olympiad 2023

April 14, 2023

geometryMetric Relationperpendicularity

Problem Statement

We consider triangle

ABC

with

\angle BAC = 90^{\circ}

and

\angle ABC = 60^{\circ}.

Let

D \in (AC) , E \in (AB),

such that

CD = 2 \cdot DA

and

DE

is bisector of

\angle ADB.

Denote by

M

the intersection of

CE

and

BD

, and by

P

the intersection of

DE

and

AM

.
a) Show that

AM \perp BD

.
b) Show that

3 \cdot PB = 2 \cdot CM

.

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