2

Part of 2008 Poland - Second Round

Problems(2)

Equal angles in the pentagon and perpendicular lines

Source: Polish MO 2008 Second Round (1st day)

In the convex pentagon

ABCDE

following equalities holds: \angle ABD\equal{} \angle ACE, \angle ACB\equal{}\angle ACD, \angle ADC\equal{}\angle ADE and \angle ADB\equal{}\angle AEC. The point

S

is the intersection of the segments

BD

CE

. Prove that lines

AS

CD

are perpendicular.

geometrygeometric transformationreflectiongeometry proposed

Geometrical identity

Source: Polish Mathematical Olympiad Second Round (day 2)

We are given a triangle

ABC

such that AC \equal{} BC. There is a point

D

lying on the segment

AB

AD < DB

E

is symmetrical to

A

with respect to

CD

. Prove that: \frac {AC}{CD} \equal{} \frac {BE}{BD \minus{} AD}

geometrygeometric transformationreflectiongeometry proposed